Shared Qs (022)

1. 
   

   Question

   prevnext

   Which graph shows a 316\(^\circ\) angle in standard position?

   
   

   
   
   1. A
   2. B
   3. C
   4. D
   5. E
   6. F
   7. G
   8. H
   
   

   Solution

   Show solution

   The standard position of an angle puts the initial side pointing East (horizontally to the right). A positive angle opens COUNTERCLOCKWISE, and a negative angle opens clockwise. The terminal side of the angle is placed at the end of the angle’s arc.

   In this case the angle is positive, so the angle rotates counterclockwise. Thus, the correct answer is B.

   
   
2. 
   

   Question

   prevnext

   Which graph shows a -181\(^\circ\) angle in standard position?

   
   

   
   
   1. A
   2. B
   3. C
   4. D
   5. E
   6. F
   7. G
   8. H
   
   

   Solution

   Show solution

   The standard position of an angle puts the initial side pointing East (horizontally to the right). A positive angle opens COUNTERCLOCKWISE, and a negative angle opens clockwise. The terminal side of the angle is placed at the end of the angle’s arc.

   In this case the angle is negative, so the angle rotates clockwise. Thus, the correct answer is D.

   
   
3. 
   

   Question

   prevnext

   Which graph shows a 155\(^\circ\) angle in standard position?

   
   

   
   
   1. A
   2. B
   3. C
   4. D
   5. E
   6. F
   7. G
   8. H
   
   

   Solution

   Show solution

   The standard position of an angle puts the initial side pointing East (horizontally to the right). A positive angle opens COUNTERCLOCKWISE, and a negative angle opens clockwise. The terminal side of the angle is placed at the end of the angle’s arc.

   In this case the angle is positive, so the angle rotates counterclockwise. Thus, the correct answer is C.

   
   
4. 
   

   Question

   prevnext

   Which graph shows a -77\(^\circ\) angle in standard position?

   
   

   
   
   1. A
   2. B
   3. C
   4. D
   5. E
   6. F
   7. G
   8. H
   
   

   Solution

   Show solution

   The standard position of an angle puts the initial side pointing East (horizontally to the right). A positive angle opens COUNTERCLOCKWISE, and a negative angle opens clockwise. The terminal side of the angle is placed at the end of the angle’s arc.

   In this case the angle is negative, so the angle rotates clockwise. Thus, the correct answer is B.

   
   
5. 
   

   Question

   prevnext

   Which graph shows a -172\(^\circ\) angle in standard position?

   
   

   
   
   1. A
   2. B
   3. C
   4. D
   5. E
   6. F
   7. G
   8. H
   
   

   Solution

   Show solution

   The standard position of an angle puts the initial side pointing East (horizontally to the right). A positive angle opens COUNTERCLOCKWISE, and a negative angle opens clockwise. The terminal side of the angle is placed at the end of the angle’s arc.

   In this case the angle is negative, so the angle rotates clockwise. Thus, the correct answer is A.

   
   
6. 
   

   Question

   prevnext

   Which graph shows a -163\(^\circ\) angle in standard position?

   
   

   
   
   1. A
   2. B
   3. C
   4. D
   5. E
   6. F
   7. G
   8. H
   
   

   Solution

   Show solution

   The standard position of an angle puts the initial side pointing East (horizontally to the right). A positive angle opens COUNTERCLOCKWISE, and a negative angle opens clockwise. The terminal side of the angle is placed at the end of the angle’s arc.

   In this case the angle is negative, so the angle rotates clockwise. Thus, the correct answer is E.

   
   
7. 
   

   Question

   prevnext

   Which graph shows a 86\(^\circ\) angle in standard position?

   
   

   
   
   1. A
   2. B
   3. C
   4. D
   5. E
   6. F
   7. G
   8. H
   
   

   Solution

   Show solution

   The standard position of an angle puts the initial side pointing East (horizontally to the right). A positive angle opens COUNTERCLOCKWISE, and a negative angle opens clockwise. The terminal side of the angle is placed at the end of the angle’s arc.

   In this case the angle is positive, so the angle rotates counterclockwise. Thus, the correct answer is G.

   
   
8. 
   

   Question

   prevnext

   Which graph shows a -224\(^\circ\) angle in standard position?

   
   

   
   
   1. A
   2. B
   3. C
   4. D
   5. E
   6. F
   7. G
   8. H
   
   

   Solution

   Show solution

   The standard position of an angle puts the initial side pointing East (horizontally to the right). A positive angle opens COUNTERCLOCKWISE, and a negative angle opens clockwise. The terminal side of the angle is placed at the end of the angle’s arc.

   In this case the angle is negative, so the angle rotates clockwise. Thus, the correct answer is F.

   
   
9. 
   

   Question

   prevnext

   Which graph shows a 47\(^\circ\) angle in standard position?

   
   

   
   
   1. A
   2. B
   3. C
   4. D
   5. E
   6. F
   7. G
   8. H
   
   

   Solution

   Show solution

   The standard position of an angle puts the initial side pointing East (horizontally to the right). A positive angle opens COUNTERCLOCKWISE, and a negative angle opens clockwise. The terminal side of the angle is placed at the end of the angle’s arc.

   In this case the angle is positive, so the angle rotates counterclockwise. Thus, the correct answer is F.

   
   
10. 
    

    Question

    prevnext

    Which graph shows a 87\(^\circ\) angle in standard position?

    
    

    
    
    1. A
    2. B
    3. C
    4. D
    5. E
    6. F
    7. G
    8. H
    
    

    Solution

    Show solution

    The standard position of an angle puts the initial side pointing East (horizontally to the right). A positive angle opens COUNTERCLOCKWISE, and a negative angle opens clockwise. The terminal side of the angle is placed at the end of the angle’s arc.

    In this case the angle is positive, so the angle rotates counterclockwise. Thus, the correct answer is C.

    
    
11. 
    

    Question

    prevnext

    Which graph shows a -1384\(^\circ\) angle in standard position?

    
    

    
    
    1. A
    2. B
    3. C
    4. D
    5. E
    6. F
    7. G
    8. H
    
    

    Solution

    Show solution

    The given angle is -1384\(^\circ\). The angle is negative, so the angle rotates clockwise.

    Notice if we divide 1384 by 360 we get 3 with a remainder of 304. So, rotate 3 full times and then also rotate by 304\(^\circ\).

    Notice that 304\(^\circ\) is between 270\(^\circ\) and 360\(^\circ\), so it’ll need between 3 and 4 quarter turns after the 3 full turns (all clockwise). That puts the terminal side in quadrant 1 (quadrant I).

    
    

    
    
12. 
    

    Question

    prevnext

    Which graph shows a 425\(^\circ\) angle in standard position?

    
    

    
    
    1. A
    2. B
    3. C
    4. D
    5. E
    6. F
    7. G
    8. H
    
    

    Solution

    Show solution

    The given angle is 425\(^\circ\). The angle is positive, so the angle rotates counterclockwise.

    Notice if we divide 425 by 360 we get 1 with a remainder of 65. So, rotate 1 full times and then also rotate by 65\(^\circ\).

    Notice that 65\(^\circ\) is between 0\(^\circ\) and 90\(^\circ\), so it’ll need between 0 and 1 quarter turns after the 1 full turns (all counterclockwise). That puts the terminal side in quadrant 1 (quadrant I).

    
    

    
    
13. 
    

    Question

    prevnext

    Which graph shows a -414\(^\circ\) angle in standard position?

    
    

    
    
    1. A
    2. B
    3. C
    4. D
    5. E
    6. F
    7. G
    8. H
    
    

    Solution

    Show solution

    The given angle is -414\(^\circ\). The angle is negative, so the angle rotates clockwise.

    Notice if we divide 414 by 360 we get 1 with a remainder of 54. So, rotate 1 full times and then also rotate by 54\(^\circ\).

    Notice that 54\(^\circ\) is between 0\(^\circ\) and 90\(^\circ\), so it’ll need between 0 and 1 quarter turns after the 1 full turns (all clockwise). That puts the terminal side in quadrant 4 (quadrant IV).

    
    

    
    
14. 
    

    Question

    prevnext

    Which graph shows a -673\(^\circ\) angle in standard position?

    
    

    
    
    1. A
    2. B
    3. C
    4. D
    5. E
    6. F
    7. G
    8. H
    
    

    Solution

    Show solution

    The given angle is -673\(^\circ\). The angle is negative, so the angle rotates clockwise.

    Notice if we divide 673 by 360 we get 1 with a remainder of 313. So, rotate 1 full times and then also rotate by 313\(^\circ\).

    Notice that 313\(^\circ\) is between 270\(^\circ\) and 360\(^\circ\), so it’ll need between 3 and 4 quarter turns after the 1 full turns (all clockwise). That puts the terminal side in quadrant 1 (quadrant I).

    
    

    
    
15. 
    

    Question

    prevnext

    Which graph shows a 1324\(^\circ\) angle in standard position?

    
    

    
    
    1. A
    2. B
    3. C
    4. D
    5. E
    6. F
    7. G
    8. H
    
    

    Solution

    Show solution

    The given angle is 1324\(^\circ\). The angle is positive, so the angle rotates counterclockwise.

    Notice if we divide 1324 by 360 we get 3 with a remainder of 244. So, rotate 3 full times and then also rotate by 244\(^\circ\).

    Notice that 244\(^\circ\) is between 180\(^\circ\) and 270\(^\circ\), so it’ll need between 2 and 3 quarter turns after the 3 full turns (all counterclockwise). That puts the terminal side in quadrant 3 (quadrant III).

    
    

    
    
16. 
    

    Question

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    Which graph shows a -1035\(^\circ\) angle in standard position?

    
    

    
    
    1. A
    2. B
    3. C
    4. D
    5. E
    6. F
    7. G
    8. H
    
    

    Solution

    Show solution

    The given angle is -1035\(^\circ\). The angle is negative, so the angle rotates clockwise.

    Notice if we divide 1035 by 360 we get 2 with a remainder of 315. So, rotate 2 full times and then also rotate by 315\(^\circ\).

    Notice that 315\(^\circ\) is between 270\(^\circ\) and 360\(^\circ\), so it’ll need between 3 and 4 quarter turns after the 2 full turns (all clockwise). That puts the terminal side in quadrant 1 (quadrant I).

    
    

    
    
17. 
    

    Question

    prevnext

    Which graph shows a -1364\(^\circ\) angle in standard position?

    
    

    
    
    1. A
    2. B
    3. C
    4. D
    5. E
    6. F
    7. G
    8. H
    
    

    Solution

    Show solution

    The given angle is -1364\(^\circ\). The angle is negative, so the angle rotates clockwise.

    Notice if we divide 1364 by 360 we get 3 with a remainder of 284. So, rotate 3 full times and then also rotate by 284\(^\circ\).

    Notice that 284\(^\circ\) is between 270\(^\circ\) and 360\(^\circ\), so it’ll need between 3 and 4 quarter turns after the 3 full turns (all clockwise). That puts the terminal side in quadrant 1 (quadrant I).

    
    

    
    
18. 
    

    Question

    prevnext

    Which graph shows a 517\(^\circ\) angle in standard position?

    
    

    
    
    1. A
    2. B
    3. C
    4. D
    5. E
    6. F
    7. G
    8. H
    
    

    Solution

    Show solution

    The given angle is 517\(^\circ\). The angle is positive, so the angle rotates counterclockwise.

    Notice if we divide 517 by 360 we get 1 with a remainder of 157. So, rotate 1 full times and then also rotate by 157\(^\circ\).

    Notice that 157\(^\circ\) is between 90\(^\circ\) and 180\(^\circ\), so it’ll need between 1 and 2 quarter turns after the 1 full turns (all counterclockwise). That puts the terminal side in quadrant 2 (quadrant II).

    
    

    
    
19. 
    

    Question

    prevnext

    Which graph shows a 1057\(^\circ\) angle in standard position?

    
    

    
    
    1. A
    2. B
    3. C
    4. D
    5. E
    6. F
    7. G
    8. H
    
    

    Solution

    Show solution

    The given angle is 1057\(^\circ\). The angle is positive, so the angle rotates counterclockwise.

    Notice if we divide 1057 by 360 we get 2 with a remainder of 337. So, rotate 2 full times and then also rotate by 337\(^\circ\).

    Notice that 337\(^\circ\) is between 270\(^\circ\) and 360\(^\circ\), so it’ll need between 3 and 4 quarter turns after the 2 full turns (all counterclockwise). That puts the terminal side in quadrant 4 (quadrant IV).

    
    

    
    
20. 
    

    Question

    prevnext

    Which graph shows a -1023\(^\circ\) angle in standard position?

    
    

    
    
    1. A
    2. B
    3. C
    4. D
    5. E
    6. F
    7. G
    8. H
    
    

    Solution

    Show solution

    The given angle is -1023\(^\circ\). The angle is negative, so the angle rotates clockwise.

    Notice if we divide 1023 by 360 we get 2 with a remainder of 303. So, rotate 2 full times and then also rotate by 303\(^\circ\).

    Notice that 303\(^\circ\) is between 270\(^\circ\) and 360\(^\circ\), so it’ll need between 3 and 4 quarter turns after the 2 full turns (all clockwise). That puts the terminal side in quadrant 1 (quadrant I).

    
    

    
    
21. 
    

    Question

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    Find an angle measure (in degrees) between 0\(^\circ\) and 360\(^\circ\) that is coterminal to an angle of -789\(^\circ\).

    
    

    Solution

    Show solution

    The given angle, -789\(^\circ\), is less than 0\(^\circ\), so add 360\(^\circ\) until you get an angle between 0\(^\circ\) and 360\(^\circ\).

    \[-789+360=-429\] \[-429+360=-69\] \[-69+360=291\]

    Thus, the answer is 291\(^\circ\). We can graph both angles: -789\(^\circ\) and 291\(^\circ\).

    
    

    Notice both terminal sides are in the same spot, even though the rotations to get there are different.

    
    
22. 
    

    Question

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    Find an angle measure (in degrees) between 0\(^\circ\) and 360\(^\circ\) that is coterminal to an angle of -952\(^\circ\).

    
    

    Solution

    Show solution

    The given angle, -952\(^\circ\), is less than 0\(^\circ\), so add 360\(^\circ\) until you get an angle between 0\(^\circ\) and 360\(^\circ\).

    \[-952+360=-592\] \[-592+360=-232\] \[-232+360=128\]

    Thus, the answer is 128\(^\circ\). We can graph both angles: -952\(^\circ\) and 128\(^\circ\).

    
    

    Notice both terminal sides are in the same spot, even though the rotations to get there are different.

    
    
23. 
    

    Question

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    Find an angle measure (in degrees) between 0\(^\circ\) and 360\(^\circ\) that is coterminal to an angle of 786\(^\circ\).

    
    

    Solution

    Show solution

    The given angle, 786\(^\circ\), is more than 360\(^\circ\), so subtract 360\(^\circ\) until you get an angle between 0\(^\circ\) and 360\(^\circ\).

    \[786-360=426\] \[426-360=66\]

    Thus, the answer is 66\(^\circ\). We can graph both angles: 786\(^\circ\) and 66\(^\circ\).

    
    

    Notice both terminal sides are in the same spot, even though the rotations to get there are different.

    
    
24. 
    

    Question

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    Find an angle measure (in degrees) between 0\(^\circ\) and 360\(^\circ\) that is coterminal to an angle of 1256\(^\circ\).

    
    

    Solution

    Show solution

    The given angle, 1256\(^\circ\), is more than 360\(^\circ\), so subtract 360\(^\circ\) until you get an angle between 0\(^\circ\) and 360\(^\circ\).

    \[1256-360=896\] \[896-360=536\] \[536-360=176\]

    Thus, the answer is 176\(^\circ\). We can graph both angles: 1256\(^\circ\) and 176\(^\circ\).

    
    

    Notice both terminal sides are in the same spot, even though the rotations to get there are different.

    
    
25. 
    

    Question

    prevnext

    Find an angle measure (in degrees) between 0\(^\circ\) and 360\(^\circ\) that is coterminal to an angle of -1424\(^\circ\).

    
    

    Solution

    Show solution

    The given angle, -1424\(^\circ\), is less than 0\(^\circ\), so add 360\(^\circ\) until you get an angle between 0\(^\circ\) and 360\(^\circ\).

    \[-1424+360=-1064\] \[-1064+360=-704\] \[-704+360=-344\] \[-344+360=16\]

    Thus, the answer is 16\(^\circ\). We can graph both angles: -1424\(^\circ\) and 16\(^\circ\).

    
    

    Notice both terminal sides are in the same spot, even though the rotations to get there are different.

    
    
26. 
    

    Question

    prevnext

    Find an angle measure (in degrees) between 0\(^\circ\) and 360\(^\circ\) that is coterminal to an angle of 1124\(^\circ\).

    
    

    Solution

    Show solution

    The given angle, 1124\(^\circ\), is more than 360\(^\circ\), so subtract 360\(^\circ\) until you get an angle between 0\(^\circ\) and 360\(^\circ\).

    \[1124-360=764\] \[764-360=404\] \[404-360=44\]

    Thus, the answer is 44\(^\circ\). We can graph both angles: 1124\(^\circ\) and 44\(^\circ\).

    
    

    Notice both terminal sides are in the same spot, even though the rotations to get there are different.

    
    
27. 
    

    Question

    prevnext

    Find an angle measure (in degrees) between 0\(^\circ\) and 360\(^\circ\) that is coterminal to an angle of 1116\(^\circ\).

    
    

    Solution

    Show solution

    The given angle, 1116\(^\circ\), is more than 360\(^\circ\), so subtract 360\(^\circ\) until you get an angle between 0\(^\circ\) and 360\(^\circ\).

    \[1116-360=756\] \[756-360=396\] \[396-360=36\]

    Thus, the answer is 36\(^\circ\). We can graph both angles: 1116\(^\circ\) and 36\(^\circ\).

    
    

    Notice both terminal sides are in the same spot, even though the rotations to get there are different.

    
    
28. 
    

    Question

    prevnext

    Find an angle measure (in degrees) between 0\(^\circ\) and 360\(^\circ\) that is coterminal to an angle of -957\(^\circ\).

    
    

    Solution

    Show solution

    The given angle, -957\(^\circ\), is less than 0\(^\circ\), so add 360\(^\circ\) until you get an angle between 0\(^\circ\) and 360\(^\circ\).

    \[-957+360=-597\] \[-597+360=-237\] \[-237+360=123\]

    Thus, the answer is 123\(^\circ\). We can graph both angles: -957\(^\circ\) and 123\(^\circ\).

    
    

    Notice both terminal sides are in the same spot, even though the rotations to get there are different.

    
    
29. 
    

    Question

    prevnext

    Find an angle measure (in degrees) between 0\(^\circ\) and 360\(^\circ\) that is coterminal to an angle of 1241\(^\circ\).

    
    

    Solution

    Show solution

    The given angle, 1241\(^\circ\), is more than 360\(^\circ\), so subtract 360\(^\circ\) until you get an angle between 0\(^\circ\) and 360\(^\circ\).

    \[1241-360=881\] \[881-360=521\] \[521-360=161\]

    Thus, the answer is 161\(^\circ\). We can graph both angles: 1241\(^\circ\) and 161\(^\circ\).

    
    

    Notice both terminal sides are in the same spot, even though the rotations to get there are different.

    
    
30. 
    

    Question

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    Find an angle measure (in degrees) between 0\(^\circ\) and 360\(^\circ\) that is coterminal to an angle of 411\(^\circ\).

    
    

    Solution

    Show solution

    The given angle, 411\(^\circ\), is more than 360\(^\circ\), so subtract 360\(^\circ\) until you get an angle between 0\(^\circ\) and 360\(^\circ\).

    \[411-360=51\]

    Thus, the answer is 51\(^\circ\). We can graph both angles: 411\(^\circ\) and 51\(^\circ\).

    
    

    Notice both terminal sides are in the same spot, even though the rotations to get there are different.

    
    
31. 
    

    Question

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    Convert 1728\(^\circ\) to rotations. How many rotations is equivalent to 1728 degrees?

    
    

    Solution

    Show solution

    One rotation is equivalent to 360\(^\circ\). Use this equivalence as a conversion factor.

    \[1728^\circ \cdot \frac{1~\mathrm{ rotation}}{360^\circ} ~~=~~ 4.8~\mathrm{ rotations}\]

    
    

    
    
32. 
    

    Question

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    Convert 1152\(^\circ\) to rotations. How many rotations is equivalent to 1152 degrees?

    
    

    Solution

    Show solution

    One rotation is equivalent to 360\(^\circ\). Use this equivalence as a conversion factor.

    \[1152^\circ \cdot \frac{1~\mathrm{ rotation}}{360^\circ} ~~=~~ 3.2~\mathrm{ rotations}\]

    
    

    
    
33. 
    

    Question

    prevnext

    Convert 2988\(^\circ\) to rotations. How many rotations is equivalent to 2988 degrees?

    
    

    Solution

    Show solution

    One rotation is equivalent to 360\(^\circ\). Use this equivalence as a conversion factor.

    \[2988^\circ \cdot \frac{1~\mathrm{ rotation}}{360^\circ} ~~=~~ 8.3~\mathrm{ rotations}\]

    
    

    
    
34. 
    

    Question

    prevnext

    Convert 1224\(^\circ\) to rotations. How many rotations is equivalent to 1224 degrees?

    
    

    Solution

    Show solution

    One rotation is equivalent to 360\(^\circ\). Use this equivalence as a conversion factor.

    \[1224^\circ \cdot \frac{1~\mathrm{ rotation}}{360^\circ} ~~=~~ 3.4~\mathrm{ rotations}\]

    
    

    
    
35. 
    

    Question

    prevnext

    Convert 324\(^\circ\) to rotations. How many rotations is equivalent to 324 degrees?

    
    

    Solution

    Show solution

    One rotation is equivalent to 360\(^\circ\). Use this equivalence as a conversion factor.

    \[324^\circ \cdot \frac{1~\mathrm{ rotation}}{360^\circ} ~~=~~ 0.9~\mathrm{ rotations}\]

    
    

    
    
36. 
    

    Question

    prevnext

    Convert 324\(^\circ\) to rotations. How many rotations is equivalent to 324 degrees?

    
    

    Solution

    Show solution

    One rotation is equivalent to 360\(^\circ\). Use this equivalence as a conversion factor.

    \[324^\circ \cdot \frac{1~\mathrm{ rotation}}{360^\circ} ~~=~~ 0.9~\mathrm{ rotations}\]

    
    

    
    
37. 
    

    Question

    prevnext

    Convert 2844\(^\circ\) to rotations. How many rotations is equivalent to 2844 degrees?

    
    

    Solution

    Show solution

    One rotation is equivalent to 360\(^\circ\). Use this equivalence as a conversion factor.

    \[2844^\circ \cdot \frac{1~\mathrm{ rotation}}{360^\circ} ~~=~~ 7.9~\mathrm{ rotations}\]

    
    

    
    
38. 
    

    Question

    prevnext

    Convert 2916\(^\circ\) to rotations. How many rotations is equivalent to 2916 degrees?

    
    

    Solution

    Show solution

    One rotation is equivalent to 360\(^\circ\). Use this equivalence as a conversion factor.

    \[2916^\circ \cdot \frac{1~\mathrm{ rotation}}{360^\circ} ~~=~~ 8.1~\mathrm{ rotations}\]

    
    

    
    
39. 
    

    Question

    prevnext

    Convert 684\(^\circ\) to rotations. How many rotations is equivalent to 684 degrees?

    
    

    Solution

    Show solution

    One rotation is equivalent to 360\(^\circ\). Use this equivalence as a conversion factor.

    \[684^\circ \cdot \frac{1~\mathrm{ rotation}}{360^\circ} ~~=~~ 1.9~\mathrm{ rotations}\]

    
    

    
    
40. 
    

    Question

    prevnext

    Convert 3276\(^\circ\) to rotations. How many rotations is equivalent to 3276 degrees?

    
    

    Solution

    Show solution

    One rotation is equivalent to 360\(^\circ\). Use this equivalence as a conversion factor.

    \[3276^\circ \cdot \frac{1~\mathrm{ rotation}}{360^\circ} ~~=~~ 9.1~\mathrm{ rotations}\]

    
    

    
    
41. 
    

    Question

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    Convert 8.1 rotations to degrees. How many degrees is equivalent to 8.1 rotations?

    
    

    Solution

    Show solution

    One rotation is equivalent to 360\(^\circ\). Use this equivalence as a conversion factor.

    \[8.1~\mathrm{ rotations} \cdot \frac{360^\circ}{1~\mathrm{ rotation}} ~~=~~ 2916^\circ\]

    
    

    
    
42. 
    

    Question

    prevnext

    Convert 6.9 rotations to degrees. How many degrees is equivalent to 6.9 rotations?

    
    

    Solution

    Show solution

    One rotation is equivalent to 360\(^\circ\). Use this equivalence as a conversion factor.

    \[6.9~\mathrm{ rotations} \cdot \frac{360^\circ}{1~\mathrm{ rotation}} ~~=~~ 2484^\circ\]

    
    

    
    
43. 
    

    Question

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    Convert 4.3 rotations to degrees. How many degrees is equivalent to 4.3 rotations?

    
    

    Solution

    Show solution

    One rotation is equivalent to 360\(^\circ\). Use this equivalence as a conversion factor.

    \[4.3~\mathrm{ rotations} \cdot \frac{360^\circ}{1~\mathrm{ rotation}} ~~=~~ 1548^\circ\]

    
    

    
    
44. 
    

    Question

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    Convert 1.6 rotations to degrees. How many degrees is equivalent to 1.6 rotations?

    
    

    Solution

    Show solution

    One rotation is equivalent to 360\(^\circ\). Use this equivalence as a conversion factor.

    \[1.6~\mathrm{ rotations} \cdot \frac{360^\circ}{1~\mathrm{ rotation}} ~~=~~ 576^\circ\]

    
    

    
    
45. 
    

    Question

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    Convert 7.4 rotations to degrees. How many degrees is equivalent to 7.4 rotations?

    
    

    Solution

    Show solution

    One rotation is equivalent to 360\(^\circ\). Use this equivalence as a conversion factor.

    \[7.4~\mathrm{ rotations} \cdot \frac{360^\circ}{1~\mathrm{ rotation}} ~~=~~ 2664^\circ\]

    
    

    
    
46. 
    

    Question

    prevnext

    Convert 9.6 rotations to degrees. How many degrees is equivalent to 9.6 rotations?

    
    

    Solution

    Show solution

    One rotation is equivalent to 360\(^\circ\). Use this equivalence as a conversion factor.

    \[9.6~\mathrm{ rotations} \cdot \frac{360^\circ}{1~\mathrm{ rotation}} ~~=~~ 3456^\circ\]

    
    

    
    
47. 
    

    Question

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    Convert 8.7 rotations to degrees. How many degrees is equivalent to 8.7 rotations?

    
    

    Solution

    Show solution

    One rotation is equivalent to 360\(^\circ\). Use this equivalence as a conversion factor.

    \[8.7~\mathrm{ rotations} \cdot \frac{360^\circ}{1~\mathrm{ rotation}} ~~=~~ 3132^\circ\]

    
    

    
    
48. 
    

    Question

    prevnext

    Convert 3.8 rotations to degrees. How many degrees is equivalent to 3.8 rotations?

    
    

    Solution

    Show solution

    One rotation is equivalent to 360\(^\circ\). Use this equivalence as a conversion factor.

    \[3.8~\mathrm{ rotations} \cdot \frac{360^\circ}{1~\mathrm{ rotation}} ~~=~~ 1368^\circ\]

    
    

    
    
49. 
    

    Question

    prevnext

    Convert 9.5 rotations to degrees. How many degrees is equivalent to 9.5 rotations?

    
    

    Solution

    Show solution

    One rotation is equivalent to 360\(^\circ\). Use this equivalence as a conversion factor.

    \[9.5~\mathrm{ rotations} \cdot \frac{360^\circ}{1~\mathrm{ rotation}} ~~=~~ 3420^\circ\]

    
    

    
    
50. 
    

    Question

    prevnext

    Convert 0.8 rotations to degrees. How many degrees is equivalent to 0.8 rotations?

    
    

    Solution

    Show solution

    One rotation is equivalent to 360\(^\circ\). Use this equivalence as a conversion factor.

    \[0.8~\mathrm{ rotations} \cdot \frac{360^\circ}{1~\mathrm{ rotation}} ~~=~~ 288^\circ\]

    
    

    
    
51. 
    

    Question

    prevnext

    Convert 211\(^\circ\) to radians. How many radians is equivalent to 211 degrees?

    The tolerance is \(\pm 0.01\) radians.

    
    

    Solution

    Show solution

    One full rotation is equivalent to 360\(^\circ\), and one full rotation is also equivalent to \(2\pi\) radians. Use this equivalence as a conversion factor.

    \[211^\circ \cdot \frac{2 \pi ~\mathrm{rad}}{360^\circ} ~~=~~ 3.6826447~\mathrm{ rad}\]

    
    

    
    
52. 
    

    Question

    prevnext

    Convert 141\(^\circ\) to radians. How many radians is equivalent to 141 degrees?

    The tolerance is \(\pm 0.01\) radians.

    
    

    Solution

    Show solution

    One full rotation is equivalent to 360\(^\circ\), and one full rotation is also equivalent to \(2\pi\) radians. Use this equivalence as a conversion factor.

    \[141^\circ \cdot \frac{2 \pi ~\mathrm{rad}}{360^\circ} ~~=~~ 2.4609142~\mathrm{ rad}\]

    
    

    
    
53. 
    

    Question

    prevnext

    Convert 34\(^\circ\) to radians. How many radians is equivalent to 34 degrees?

    The tolerance is \(\pm 0.01\) radians.

    
    

    Solution

    Show solution

    One full rotation is equivalent to 360\(^\circ\), and one full rotation is also equivalent to \(2\pi\) radians. Use this equivalence as a conversion factor.

    \[34^\circ \cdot \frac{2 \pi ~\mathrm{rad}}{360^\circ} ~~=~~ 0.5934119~\mathrm{ rad}\]

    
    

    
    
54. 
    

    Question

    prevnext

    Convert 34\(^\circ\) to radians. How many radians is equivalent to 34 degrees?

    The tolerance is \(\pm 0.01\) radians.

    
    

    Solution

    Show solution

    One full rotation is equivalent to 360\(^\circ\), and one full rotation is also equivalent to \(2\pi\) radians. Use this equivalence as a conversion factor.

    \[34^\circ \cdot \frac{2 \pi ~\mathrm{rad}}{360^\circ} ~~=~~ 0.5934119~\mathrm{ rad}\]

    
    

    
    
55. 
    

    Question

    prevnext

    Convert 125\(^\circ\) to radians. How many radians is equivalent to 125 degrees?

    The tolerance is \(\pm 0.01\) radians.

    
    

    Solution

    Show solution

    One full rotation is equivalent to 360\(^\circ\), and one full rotation is also equivalent to \(2\pi\) radians. Use this equivalence as a conversion factor.

    \[125^\circ \cdot \frac{2 \pi ~\mathrm{rad}}{360^\circ} ~~=~~ 2.1816616~\mathrm{ rad}\]

    
    

    
    
56. 
    

    Question

    prevnext

    Convert 349\(^\circ\) to radians. How many radians is equivalent to 349 degrees?

    The tolerance is \(\pm 0.01\) radians.

    
    

    Solution

    Show solution

    One full rotation is equivalent to 360\(^\circ\), and one full rotation is also equivalent to \(2\pi\) radians. Use this equivalence as a conversion factor.

    \[349^\circ \cdot \frac{2 \pi ~\mathrm{rad}}{360^\circ} ~~=~~ 6.0911991~\mathrm{ rad}\]

    
    

    
    
57. 
    

    Question

    prevnext

    Convert 194\(^\circ\) to radians. How many radians is equivalent to 194 degrees?

    The tolerance is \(\pm 0.01\) radians.

    
    

    Solution

    Show solution

    One full rotation is equivalent to 360\(^\circ\), and one full rotation is also equivalent to \(2\pi\) radians. Use this equivalence as a conversion factor.

    \[194^\circ \cdot \frac{2 \pi ~\mathrm{rad}}{360^\circ} ~~=~~ 3.3859387~\mathrm{ rad}\]

    
    

    
    
58. 
    

    Question

    prevnext

    Convert 195\(^\circ\) to radians. How many radians is equivalent to 195 degrees?

    The tolerance is \(\pm 0.01\) radians.

    
    

    Solution

    Show solution

    One full rotation is equivalent to 360\(^\circ\), and one full rotation is also equivalent to \(2\pi\) radians. Use this equivalence as a conversion factor.

    \[195^\circ \cdot \frac{2 \pi ~\mathrm{rad}}{360^\circ} ~~=~~ 3.403392~\mathrm{ rad}\]

    
    

    
    
59. 
    

    Question

    prevnext

    Convert 202\(^\circ\) to radians. How many radians is equivalent to 202 degrees?

    The tolerance is \(\pm 0.01\) radians.

    
    

    Solution

    Show solution

    One full rotation is equivalent to 360\(^\circ\), and one full rotation is also equivalent to \(2\pi\) radians. Use this equivalence as a conversion factor.

    \[202^\circ \cdot \frac{2 \pi ~\mathrm{rad}}{360^\circ} ~~=~~ 3.5255651~\mathrm{ rad}\]

    
    

    
    
60. 
    

    Question

    prevnext

    Convert 345\(^\circ\) to radians. How many radians is equivalent to 345 degrees?

    The tolerance is \(\pm 0.01\) radians.

    
    

    Solution

    Show solution

    One full rotation is equivalent to 360\(^\circ\), and one full rotation is also equivalent to \(2\pi\) radians. Use this equivalence as a conversion factor.

    \[345^\circ \cdot \frac{2 \pi ~\mathrm{rad}}{360^\circ} ~~=~~ 6.0213859~\mathrm{ rad}\]

    
    

    
    
61. 
    

    Question

    prevnext

    Convert 0.84 radians to degrees. How many degrees is equivalent to 0.84 radians?

    The tolerance is \(\pm 0.1\) degrees.

    
    

    Solution

    Show solution

    One half rotation is equivalent to 180\(^\circ\), and one half rotation is also equivalent to \(\pi\) radians. Use this equivalence as a conversion factor.

    \[0.84 ~\mathrm{rad} \cdot \frac{180^\circ}{\pi ~\mathrm{rad}} ~~=~~ 48.1284548^\circ ~~\approx~~ 48.1^\circ\]

    
    

    
    
62. 
    

    Question

    prevnext

    Convert 0.76 radians to degrees. How many degrees is equivalent to 0.76 radians?

    The tolerance is \(\pm 0.1\) degrees.

    
    

    Solution

    Show solution

    One half rotation is equivalent to 180\(^\circ\), and one half rotation is also equivalent to \(\pi\) radians. Use this equivalence as a conversion factor.

    \[0.76 ~\mathrm{rad} \cdot \frac{180^\circ}{\pi ~\mathrm{rad}} ~~=~~ 43.5447924^\circ ~~\approx~~ 43.5^\circ\]

    
    

    
    
63. 
    

    Question

    prevnext

    Convert 0.38 radians to degrees. How many degrees is equivalent to 0.38 radians?

    The tolerance is \(\pm 0.1\) degrees.

    
    

    Solution

    Show solution

    One half rotation is equivalent to 180\(^\circ\), and one half rotation is also equivalent to \(\pi\) radians. Use this equivalence as a conversion factor.

    \[0.38 ~\mathrm{rad} \cdot \frac{180^\circ}{\pi ~\mathrm{rad}} ~~=~~ 21.7723962^\circ ~~\approx~~ 21.8^\circ\]

    
    

    
    
64. 
    

    Question

    prevnext

    Convert 5.82 radians to degrees. How many degrees is equivalent to 5.82 radians?

    The tolerance is \(\pm 0.1\) degrees.

    
    

    Solution

    Show solution

    One half rotation is equivalent to 180\(^\circ\), and one half rotation is also equivalent to \(\pi\) radians. Use this equivalence as a conversion factor.

    \[5.82 ~\mathrm{rad} \cdot \frac{180^\circ}{\pi ~\mathrm{rad}} ~~=~~ 333.4614368^\circ ~~\approx~~ 333.5^\circ\]

    
    

    
    
65. 
    

    Question

    prevnext

    Convert 5.38 radians to degrees. How many degrees is equivalent to 5.38 radians?

    The tolerance is \(\pm 0.1\) degrees.

    
    

    Solution

    Show solution

    One half rotation is equivalent to 180\(^\circ\), and one half rotation is also equivalent to \(\pi\) radians. Use this equivalence as a conversion factor.

    \[5.38 ~\mathrm{rad} \cdot \frac{180^\circ}{\pi ~\mathrm{rad}} ~~=~~ 308.2512938^\circ ~~\approx~~ 308.3^\circ\]

    
    

    
    
66. 
    

    Question

    prevnext

    Convert 0.22 radians to degrees. How many degrees is equivalent to 0.22 radians?

    The tolerance is \(\pm 0.1\) degrees.

    
    

    Solution

    Show solution

    One half rotation is equivalent to 180\(^\circ\), and one half rotation is also equivalent to \(\pi\) radians. Use this equivalence as a conversion factor.

    \[0.22 ~\mathrm{rad} \cdot \frac{180^\circ}{\pi ~\mathrm{rad}} ~~=~~ 12.6050715^\circ ~~\approx~~ 12.6^\circ\]

    
    

    
    
67. 
    

    Question

    prevnext

    Convert 4.06 radians to degrees. How many degrees is equivalent to 4.06 radians?

    The tolerance is \(\pm 0.1\) degrees.

    
    

    Solution

    Show solution

    One half rotation is equivalent to 180\(^\circ\), and one half rotation is also equivalent to \(\pi\) radians. Use this equivalence as a conversion factor.

    \[4.06 ~\mathrm{rad} \cdot \frac{180^\circ}{\pi ~\mathrm{rad}} ~~=~~ 232.6208648^\circ ~~\approx~~ 232.6^\circ\]

    
    

    
    
68. 
    

    Question

    prevnext

    Convert 0.81 radians to degrees. How many degrees is equivalent to 0.81 radians?

    The tolerance is \(\pm 0.1\) degrees.

    
    

    Solution

    Show solution

    One half rotation is equivalent to 180\(^\circ\), and one half rotation is also equivalent to \(\pi\) radians. Use this equivalence as a conversion factor.

    \[0.81 ~\mathrm{rad} \cdot \frac{180^\circ}{\pi ~\mathrm{rad}} ~~=~~ 46.4095814^\circ ~~\approx~~ 46.4^\circ\]

    
    

    
    
69. 
    

    Question

    prevnext

    Convert 3.76 radians to degrees. How many degrees is equivalent to 3.76 radians?

    The tolerance is \(\pm 0.1\) degrees.

    
    

    Solution

    Show solution

    One half rotation is equivalent to 180\(^\circ\), and one half rotation is also equivalent to \(\pi\) radians. Use this equivalence as a conversion factor.

    \[3.76 ~\mathrm{rad} \cdot \frac{180^\circ}{\pi ~\mathrm{rad}} ~~=~~ 215.432131^\circ ~~\approx~~ 215.4^\circ\]

    
    

    
    
70. 
    

    Question

    prevnext

    Convert 1.8 radians to degrees. How many degrees is equivalent to 1.8 radians?

    The tolerance is \(\pm 0.1\) degrees.

    
    

    Solution

    Show solution

    One half rotation is equivalent to 180\(^\circ\), and one half rotation is also equivalent to \(\pi\) radians. Use this equivalence as a conversion factor.

    \[1.8 ~\mathrm{rad} \cdot \frac{180^\circ}{\pi ~\mathrm{rad}} ~~=~~ 103.1324031^\circ ~~\approx~~ 103.1^\circ\]

    
    

    
    
71. 
    

    Question

    prevnext

    An arc is drawn along the circumference of a circle with a radius of 7 meters. The arc’s angle is 336 degrees. What is the arc’s length in meters?

    
    

    The tolerance is \(\pm0.01\) meters.

    
    

    Solution

    Show solution

    For reference, check out the wikipedia article on central angles.

    First, find the total circumference. \[C = 2\pi r \]

    \[C = 2\pi\cdot7\]

    \[C = 14\pi\]

    Next, determine the fraction of the circle.

    \[\frac{336}{360} = \frac{14}{15}\]

    Find the fraction of the circumference.

    \[x = \frac{14}{15} \cdot 14\pi\]

    Simplify.

    \[x = \frac{196\pi}{15}\]

    Evaluate the decimal approximation.

    \[x \approx 41.050144\] So the arc length is approximately 41.050144 meters.

    
    
72. 
    

    Question

    prevnext

    An arc is drawn along the circumference of a circle with a radius of 7 meters. The arc’s angle is 320 degrees. What is the arc’s length in meters?

    
    

    The tolerance is \(\pm0.01\) meters.

    
    

    Solution

    Show solution

    For reference, check out the wikipedia article on central angles.

    First, find the total circumference. \[C = 2\pi r \]

    \[C = 2\pi\cdot7\]

    \[C = 14\pi\]

    Next, determine the fraction of the circle.

    \[\frac{320}{360} = \frac{8}{9}\]

    Find the fraction of the circumference.

    \[x = \frac{8}{9} \cdot 14\pi\]

    Simplify.

    \[x = \frac{112\pi}{9}\]

    Evaluate the decimal approximation.

    \[x \approx 39.0953752\] So the arc length is approximately 39.0953752 meters.

    
    
73. 
    

    Question

    prevnext

    An arc is drawn along the circumference of a circle with a radius of 7 meters. The arc’s angle is 324 degrees. What is the arc’s length in meters?

    
    

    The tolerance is \(\pm0.01\) meters.

    
    

    Solution

    Show solution

    For reference, check out the wikipedia article on central angles.

    First, find the total circumference. \[C = 2\pi r \]

    \[C = 2\pi\cdot7\]

    \[C = 14\pi\]

    Next, determine the fraction of the circle.

    \[\frac{324}{360} = \frac{9}{10}\]

    Find the fraction of the circumference.

    \[x = \frac{9}{10} \cdot 14\pi\]

    Simplify.

    \[x = \frac{63\pi}{5}\]

    Evaluate the decimal approximation.

    \[x \approx 39.5840674\] So the arc length is approximately 39.5840674 meters.

    
    
74. 
    

    Question

    prevnext

    An arc is drawn along the circumference of a circle with a radius of 7 meters. The arc’s angle is 36 degrees. What is the arc’s length in meters?

    
    

    The tolerance is \(\pm0.01\) meters.

    
    

    Solution

    Show solution

    For reference, check out the wikipedia article on central angles.

    First, find the total circumference. \[C = 2\pi r \]

    \[C = 2\pi\cdot7\]

    \[C = 14\pi\]

    Next, determine the fraction of the circle.

    \[\frac{36}{360} = \frac{1}{10}\]

    Find the fraction of the circumference.

    \[x = \frac{1}{10} \cdot 14\pi\]

    Simplify.

    \[x = \frac{7\pi}{5}\]

    Evaluate the decimal approximation.

    \[x \approx 4.3982297\] So the arc length is approximately 4.3982297 meters.

    
    
75. 
    

    Question

    prevnext

    An arc is drawn along the circumference of a circle with a radius of 2 meters. The arc’s angle is 220 degrees. What is the arc’s length in meters?

    
    

    The tolerance is \(\pm0.01\) meters.

    
    

    Solution

    Show solution

    For reference, check out the wikipedia article on central angles.

    First, find the total circumference. \[C = 2\pi r \]

    \[C = 2\pi\cdot2\]

    \[C = 4\pi\]

    Next, determine the fraction of the circle.

    \[\frac{220}{360} = \frac{11}{18}\]

    Find the fraction of the circumference.

    \[x = \frac{11}{18} \cdot 4\pi\]

    Simplify.

    \[x = \frac{22\pi}{9}\]

    Evaluate the decimal approximation.

    \[x \approx 7.6794487\] So the arc length is approximately 7.6794487 meters.

    
    
76. 
    

    Question

    prevnext

    An arc is drawn along the circumference of a circle with a radius of 8 meters. The arc’s angle is 324 degrees. What is the arc’s length in meters?

    
    

    The tolerance is \(\pm0.01\) meters.

    
    

    Solution

    Show solution

    For reference, check out the wikipedia article on central angles.

    First, find the total circumference. \[C = 2\pi r \]

    \[C = 2\pi\cdot8\]

    \[C = 16\pi\]

    Next, determine the fraction of the circle.

    \[\frac{324}{360} = \frac{9}{10}\]

    Find the fraction of the circumference.

    \[x = \frac{9}{10} \cdot 16\pi\]

    Simplify.

    \[x = \frac{72\pi}{5}\]

    Evaluate the decimal approximation.

    \[x \approx 45.2389342\] So the arc length is approximately 45.2389342 meters.

    
    
77. 
    

    Question

    prevnext

    An arc is drawn along the circumference of a circle with a radius of 2 meters. The arc’s angle is 225 degrees. What is the arc’s length in meters?

    
    

    The tolerance is \(\pm0.01\) meters.

    
    

    Solution

    Show solution

    For reference, check out the wikipedia article on central angles.

    First, find the total circumference. \[C = 2\pi r \]

    \[C = 2\pi\cdot2\]

    \[C = 4\pi\]

    Next, determine the fraction of the circle.

    \[\frac{225}{360} = \frac{5}{8}\]

    Find the fraction of the circumference.

    \[x = \frac{5}{8} \cdot 4\pi\]

    Simplify.

    \[x = \frac{5\pi}{2}\]

    Evaluate the decimal approximation.

    \[x \approx 7.8539816\] So the arc length is approximately 7.8539816 meters.

    
    
78. 
    

    Question

    prevnext

    An arc is drawn along the circumference of a circle with a radius of 2 meters. The arc’s angle is 252 degrees. What is the arc’s length in meters?

    
    

    The tolerance is \(\pm0.01\) meters.

    
    

    Solution

    Show solution

    For reference, check out the wikipedia article on central angles.

    First, find the total circumference. \[C = 2\pi r \]

    \[C = 2\pi\cdot2\]

    \[C = 4\pi\]

    Next, determine the fraction of the circle.

    \[\frac{252}{360} = \frac{7}{10}\]

    Find the fraction of the circumference.

    \[x = \frac{7}{10} \cdot 4\pi\]

    Simplify.

    \[x = \frac{14\pi}{5}\]

    Evaluate the decimal approximation.

    \[x \approx 8.7964594\] So the arc length is approximately 8.7964594 meters.

    
    
79. 
    

    Question

    prevnext

    An arc is drawn along the circumference of a circle with a radius of 3 meters. The arc’s angle is 135 degrees. What is the arc’s length in meters?

    
    

    The tolerance is \(\pm0.01\) meters.

    
    

    Solution

    Show solution

    For reference, check out the wikipedia article on central angles.

    First, find the total circumference. \[C = 2\pi r \]

    \[C = 2\pi\cdot3\]

    \[C = 6\pi\]

    Next, determine the fraction of the circle.

    \[\frac{135}{360} = \frac{3}{8}\]

    Find the fraction of the circumference.

    \[x = \frac{3}{8} \cdot 6\pi\]

    Simplify.

    \[x = \frac{9\pi}{4}\]

    Evaluate the decimal approximation.

    \[x \approx 7.0685835\] So the arc length is approximately 7.0685835 meters.

    
    
80. 
    

    Question

    prevnext

    An arc is drawn along the circumference of a circle with a radius of 4 meters. The arc’s angle is 108 degrees. What is the arc’s length in meters?

    
    

    The tolerance is \(\pm0.01\) meters.

    
    

    Solution

    Show solution

    For reference, check out the wikipedia article on central angles.

    First, find the total circumference. \[C = 2\pi r \]

    \[C = 2\pi\cdot4\]

    \[C = 8\pi\]

    Next, determine the fraction of the circle.

    \[\frac{108}{360} = \frac{3}{10}\]

    Find the fraction of the circumference.

    \[x = \frac{3}{10} \cdot 8\pi\]

    Simplify.

    \[x = \frac{12\pi}{5}\]

    Evaluate the decimal approximation.

    \[x \approx 7.5398224\] So the arc length is approximately 7.5398224 meters.

    
    
81. 
    

    Question

    prevnext

    An arc is drawn along the circumference of a circle with a radius of 6 meters. The arc’s angle is \(\frac{28\pi}{15}\) radians. What is the arc’s length in meters?

    
    

    The tolerance is \(\pm0.01\) meters.

    
    

    Solution

    Show solution

    This problem shows a useful property of radians. To find the arc length, just multiply the angle measure (in radians) by the radius.

    \[x = \frac{28\pi}{15}\cdot 6\]

    \[x = \frac{56\pi}{5}\]

    Evaluate the decimal approximation.

    \[x \approx 35.1858377\]

    So the arc length is approximately 35.1858377 meters.

    
    
82. 
    

    Question

    prevnext

    An arc is drawn along the circumference of a circle with a radius of 8 meters. The arc’s angle is \(\frac{5\pi}{3}\) radians. What is the arc’s length in meters?

    
    

    The tolerance is \(\pm0.01\) meters.

    
    

    Solution

    Show solution

    This problem shows a useful property of radians. To find the arc length, just multiply the angle measure (in radians) by the radius.

    \[x = \frac{5\pi}{3}\cdot 8\]

    \[x = \frac{40\pi}{3}\]

    Evaluate the decimal approximation.

    \[x \approx 41.887902\]

    So the arc length is approximately 41.887902 meters.

    
    
83. 
    

    Question

    prevnext

    An arc is drawn along the circumference of a circle with a radius of 8 meters. The arc’s angle is \(\frac{6\pi}{5}\) radians. What is the arc’s length in meters?

    
    

    The tolerance is \(\pm0.01\) meters.

    
    

    Solution

    Show solution

    This problem shows a useful property of radians. To find the arc length, just multiply the angle measure (in radians) by the radius.

    \[x = \frac{6\pi}{5}\cdot 8\]

    \[x = \frac{48\pi}{5}\]

    Evaluate the decimal approximation.

    \[x \approx 30.1592895\]

    So the arc length is approximately 30.1592895 meters.

    
    
84. 
    

    Question

    prevnext

    An arc is drawn along the circumference of a circle with a radius of 5 meters. The arc’s angle is \(\frac{4\pi}{15}\) radians. What is the arc’s length in meters?

    
    

    The tolerance is \(\pm0.01\) meters.

    
    

    Solution

    Show solution

    This problem shows a useful property of radians. To find the arc length, just multiply the angle measure (in radians) by the radius.

    \[x = \frac{4\pi}{15}\cdot 5\]

    \[x = \frac{4\pi}{3}\]

    Evaluate the decimal approximation.

    \[x \approx 4.1887902\]

    So the arc length is approximately 4.1887902 meters.

    
    
85. 
    

    Question

    prevnext

    An arc is drawn along the circumference of a circle with a radius of 5 meters. The arc’s angle is \(\frac{5\pi}{9}\) radians. What is the arc’s length in meters?

    
    

    The tolerance is \(\pm0.01\) meters.

    
    

    Solution

    Show solution

    This problem shows a useful property of radians. To find the arc length, just multiply the angle measure (in radians) by the radius.

    \[x = \frac{5\pi}{9}\cdot 5\]

    \[x = \frac{25\pi}{9}\]

    Evaluate the decimal approximation.

    \[x \approx 8.7266463\]

    So the arc length is approximately 8.7266463 meters.

    
    
86. 
    

    Question

    prevnext

    An arc is drawn along the circumference of a circle with a radius of 3 meters. The arc’s angle is \(\frac{1\pi}{4}\) radians. What is the arc’s length in meters?

    
    

    The tolerance is \(\pm0.01\) meters.

    
    

    Solution

    Show solution

    This problem shows a useful property of radians. To find the arc length, just multiply the angle measure (in radians) by the radius.

    \[x = \frac{1\pi}{4}\cdot 3\]

    \[x = \frac{3\pi}{4}\]

    Evaluate the decimal approximation.

    \[x \approx 2.3561945\]

    So the arc length is approximately 2.3561945 meters.

    
    
87. 
    

    Question

    prevnext

    An arc is drawn along the circumference of a circle with a radius of 6 meters. The arc’s angle is \(\frac{9\pi}{5}\) radians. What is the arc’s length in meters?

    
    

    The tolerance is \(\pm0.01\) meters.

    
    

    Solution

    Show solution

    This problem shows a useful property of radians. To find the arc length, just multiply the angle measure (in radians) by the radius.

    \[x = \frac{9\pi}{5}\cdot 6\]

    \[x = \frac{54\pi}{5}\]

    Evaluate the decimal approximation.

    \[x \approx 33.9292007\]

    So the arc length is approximately 33.9292007 meters.

    
    
88. 
    

    Question

    prevnext

    An arc is drawn along the circumference of a circle with a radius of 4 meters. The arc’s angle is \(\frac{26\pi}{15}\) radians. What is the arc’s length in meters?

    
    

    The tolerance is \(\pm0.01\) meters.

    
    

    Solution

    Show solution

    This problem shows a useful property of radians. To find the arc length, just multiply the angle measure (in radians) by the radius.

    \[x = \frac{26\pi}{15}\cdot 4\]

    \[x = \frac{104\pi}{15}\]

    Evaluate the decimal approximation.

    \[x \approx 21.7817091\]

    So the arc length is approximately 21.7817091 meters.

    
    
89. 
    

    Question

    prevnext

    An arc is drawn along the circumference of a circle with a radius of 2 meters. The arc’s angle is \(\frac{7\pi}{5}\) radians. What is the arc’s length in meters?

    
    

    The tolerance is \(\pm0.01\) meters.

    
    

    Solution

    Show solution

    This problem shows a useful property of radians. To find the arc length, just multiply the angle measure (in radians) by the radius.

    \[x = \frac{7\pi}{5}\cdot 2\]

    \[x = \frac{14\pi}{5}\]

    Evaluate the decimal approximation.

    \[x \approx 8.7964594\]

    So the arc length is approximately 8.7964594 meters.

    
    
90. 
    

    Question

    prevnext

    An arc is drawn along the circumference of a circle with a radius of 2 meters. The arc’s angle is \(\frac{2\pi}{5}\) radians. What is the arc’s length in meters?

    
    

    The tolerance is \(\pm0.01\) meters.

    
    

    Solution

    Show solution

    This problem shows a useful property of radians. To find the arc length, just multiply the angle measure (in radians) by the radius.

    \[x = \frac{2\pi}{5}\cdot 2\]

    \[x = \frac{4\pi}{5}\]

    Evaluate the decimal approximation.

    \[x \approx 2.5132741\]

    So the arc length is approximately 2.5132741 meters.

    
    
91. 
    

    Question

    prevnext

    Which graph shows an angle measuring \(\frac{4\pi}{3}\) radians in standard position?

    
    

    
    
    1. A
    2. B
    3. C
    4. D
    5. E
    6. F
    7. G
    8. H
    
    

    Solution

    Show solution

    The given angle is \(\frac{4\pi}{3}\) radians. The angle is positive, so the angle rotates counterclockwise. I prefer to think of this as \(4\) times \(\frac{\pi}{3}\), so then I imagine the top semicircle (and bottom semicircle) broken up into 3 equally-spaced wedges.

    
    

    
    
92. 
    

    Question

    prevnext

    Which graph shows an angle measuring \(\frac{4\pi}{9}\) radians in standard position?

    
    

    
    
    1. A
    2. B
    3. C
    4. D
    5. E
    6. F
    7. G
    8. H
    
    

    Solution

    Show solution

    The given angle is \(\frac{4\pi}{9}\) radians. The angle is positive, so the angle rotates counterclockwise. I prefer to think of this as \(4\) times \(\frac{\pi}{9}\), so then I imagine the top semicircle (and bottom semicircle) broken up into 9 equally-spaced wedges.

    
    

    
    
93. 
    

    Question

    prevnext

    Which graph shows an angle measuring \(\frac{2\pi}{5}\) radians in standard position?

    
    

    
    
    1. A
    2. B
    3. C
    4. D
    5. E
    6. F
    7. G
    8. H
    
    

    Solution

    Show solution

    The given angle is \(\frac{2\pi}{5}\) radians. The angle is positive, so the angle rotates counterclockwise. I prefer to think of this as \(2\) times \(\frac{\pi}{5}\), so then I imagine the top semicircle (and bottom semicircle) broken up into 5 equally-spaced wedges.

    
    

    
    
94. 
    

    Question

    prevnext

    Which graph shows an angle measuring \(\frac{6\pi}{7}\) radians in standard position?

    
    

    
    
    1. A
    2. B
    3. C
    4. D
    5. E
    6. F
    7. G
    8. H
    
    

    Solution

    Show solution

    The given angle is \(\frac{6\pi}{7}\) radians. The angle is positive, so the angle rotates counterclockwise. I prefer to think of this as \(6\) times \(\frac{\pi}{7}\), so then I imagine the top semicircle (and bottom semicircle) broken up into 7 equally-spaced wedges.

    
    

    
    
95. 
    

    Question

    prevnext

    Which graph shows an angle measuring \(\frac{21\pi}{11}\) radians in standard position?

    
    

    
    
    1. A
    2. B
    3. C
    4. D
    5. E
    6. F
    7. G
    8. H
    
    

    Solution

    Show solution

    The given angle is \(\frac{21\pi}{11}\) radians. The angle is positive, so the angle rotates counterclockwise. I prefer to think of this as \(21\) times \(\frac{\pi}{11}\), so then I imagine the top semicircle (and bottom semicircle) broken up into 11 equally-spaced wedges.

    
    

    
    
96. 
    

    Question

    prevnext

    Which graph shows an angle measuring \(\frac{18\pi}{11}\) radians in standard position?

    
    

    
    
    1. A
    2. B
    3. C
    4. D
    5. E
    6. F
    7. G
    8. H
    
    

    Solution

    Show solution

    The given angle is \(\frac{18\pi}{11}\) radians. The angle is positive, so the angle rotates counterclockwise. I prefer to think of this as \(18\) times \(\frac{\pi}{11}\), so then I imagine the top semicircle (and bottom semicircle) broken up into 11 equally-spaced wedges.

    
    

    
    
97. 
    

    Question

    prevnext

    Which graph shows an angle measuring \(\frac{9\pi}{11}\) radians in standard position?

    
    

    
    
    1. A
    2. B
    3. C
    4. D
    5. E
    6. F
    7. G
    8. H
    
    

    Solution

    Show solution

    The given angle is \(\frac{9\pi}{11}\) radians. The angle is positive, so the angle rotates counterclockwise. I prefer to think of this as \(9\) times \(\frac{\pi}{11}\), so then I imagine the top semicircle (and bottom semicircle) broken up into 11 equally-spaced wedges.

    
    

    
    
98. 
    

    Question

    prevnext

    Which graph shows an angle measuring \(\frac{7\pi}{6}\) radians in standard position?

    
    

    
    
    1. A
    2. B
    3. C
    4. D
    5. E
    6. F
    7. G
    8. H
    
    

    Solution

    Show solution

    The given angle is \(\frac{7\pi}{6}\) radians. The angle is positive, so the angle rotates counterclockwise. I prefer to think of this as \(7\) times \(\frac{\pi}{6}\), so then I imagine the top semicircle (and bottom semicircle) broken up into 6 equally-spaced wedges.

    
    

    
    
99. 
    

    Question

    prevnext

    Which graph shows an angle measuring \(\frac{18\pi}{11}\) radians in standard position?

    
    

    
    
    1. A
    2. B
    3. C
    4. D
    5. E
    6. F
    7. G
    8. H
    
    

    Solution

    Show solution

    The given angle is \(\frac{18\pi}{11}\) radians. The angle is positive, so the angle rotates counterclockwise. I prefer to think of this as \(18\) times \(\frac{\pi}{11}\), so then I imagine the top semicircle (and bottom semicircle) broken up into 11 equally-spaced wedges.

    
    

    
    
100. 
     

     Question

     prevnext

     Which graph shows an angle measuring \(\frac{5\pi}{3}\) radians in standard position?

     
     

     
     
     1. A
     2. B
     3. C
     4. D
     5. E
     6. F
     7. G
     8. H
     
     

     Solution

     Show solution

     The given angle is \(\frac{5\pi}{3}\) radians. The angle is positive, so the angle rotates counterclockwise. I prefer to think of this as \(5\) times \(\frac{\pi}{3}\), so then I imagine the top semicircle (and bottom semicircle) broken up into 3 equally-spaced wedges.

     
     

     
     
101. 
     

     Question

     prevnext

     An angle measure, \(\theta\), equals \(\frac{-46\pi}{9}\) radians. A second angle measure, \(\phi\), is coterminal to \(\theta\), and \(0\le\phi<2\pi\). Which graph shows \(\phi\) in standard position?

     
     

     
     
     1. A
     2. B
     3. C
     4. D
     5. E
     6. F
     7. G
     8. H
     
     

     Solution

     Show solution

     The given angle measure is \(\theta=\frac{-46\pi}{9}\) radians. Because the given angle is less than 0, repeatedly add \(2\pi\) until getting a coterminal angle between 0 and \(2\pi\). It should be mentioned that \(2\pi = \frac{18\pi}{9}\), so we can add fractions with common denominators.

     \[\frac{-46\pi}{9}+\frac{18\pi}{9}=\frac{-28\pi}{9}\] \[\frac{-28\pi}{9}+\frac{18\pi}{9}=\frac{-10\pi}{9}\] \[\frac{-10\pi}{9}+\frac{18\pi}{9}=\frac{8\pi}{9}\]

     We found the coterminal angle \(\phi=\frac{8\pi}{9}\). I prefer to think of this as \(8\) times \(\frac{\pi}{9}\), so then I imagine the top semicircle (and bottom semicircle) broken up into 9 equally-spaced wedges.

     
     

     
     
102. 
     

     Question

     prevnext

     An angle measure, \(\theta\), equals \(\frac{-18\pi}{7}\) radians. A second angle measure, \(\phi\), is coterminal to \(\theta\), and \(0\le\phi<2\pi\). Which graph shows \(\phi\) in standard position?

     
     

     
     
     1. A
     2. B
     3. C
     4. D
     5. E
     6. F
     7. G
     8. H
     
     

     Solution

     Show solution

     The given angle measure is \(\theta=\frac{-18\pi}{7}\) radians. Because the given angle is less than 0, repeatedly add \(2\pi\) until getting a coterminal angle between 0 and \(2\pi\). It should be mentioned that \(2\pi = \frac{14\pi}{7}\), so we can add fractions with common denominators.

     \[\frac{-18\pi}{7}+\frac{14\pi}{7}=\frac{-4\pi}{7}\] \[\frac{-4\pi}{7}+\frac{14\pi}{7}=\frac{10\pi}{7}\]

     We found the coterminal angle \(\phi=\frac{10\pi}{7}\). I prefer to think of this as \(10\) times \(\frac{\pi}{7}\), so then I imagine the top semicircle (and bottom semicircle) broken up into 7 equally-spaced wedges.

     
     

     
     
103. 
     

     Question

     prevnext

     An angle measure, \(\theta\), equals \(\frac{-12\pi}{5}\) radians. A second angle measure, \(\phi\), is coterminal to \(\theta\), and \(0\le\phi<2\pi\). Which graph shows \(\phi\) in standard position?

     
     

     
     
     1. A
     2. B
     3. C
     4. D
     5. E
     6. F
     7. G
     8. H
     
     

     Solution

     Show solution

     The given angle measure is \(\theta=\frac{-12\pi}{5}\) radians. Because the given angle is less than 0, repeatedly add \(2\pi\) until getting a coterminal angle between 0 and \(2\pi\). It should be mentioned that \(2\pi = \frac{10\pi}{5}\), so we can add fractions with common denominators.

     \[\frac{-12\pi}{5}+\frac{10\pi}{5}=\frac{-2\pi}{5}\] \[\frac{-2\pi}{5}+\frac{10\pi}{5}=\frac{8\pi}{5}\]

     We found the coterminal angle \(\phi=\frac{8\pi}{5}\). I prefer to think of this as \(8\) times \(\frac{\pi}{5}\), so then I imagine the top semicircle (and bottom semicircle) broken up into 5 equally-spaced wedges.

     
     

     
     
104. 
     

     Question

     prevnext

     An angle measure, \(\theta\), equals \(\frac{84\pi}{11}\) radians. A second angle measure, \(\phi\), is coterminal to \(\theta\), and \(0\le\phi<2\pi\). Which graph shows \(\phi\) in standard position?

     
     

     
     
     1. A
     2. B
     3. C
     4. D
     5. E
     6. F
     7. G
     8. H
     
     

     Solution

     Show solution

     The given angle measure is \(\theta=\frac{84\pi}{11}\) radians. Because the given angle is more than \(2\pi\), repeatedly subtract \(2\pi\) until getting a coterminal angle between 0 and \(2\pi\). It should be mentioned that \(2\pi = \frac{22\pi}{11}\), so we can add fractions with common denominators.

     \[\frac{84\pi}{11}-\frac{22\pi}{11}=\frac{62\pi}{11}\] \[\frac{62\pi}{11}-\frac{22\pi}{11}=\frac{40\pi}{11}\] \[\frac{40\pi}{11}-\frac{22\pi}{11}=\frac{18\pi}{11}\]

     We found the coterminal angle \(\phi=\frac{18\pi}{11}\). I prefer to think of this as \(18\) times \(\frac{\pi}{11}\), so then I imagine the top semicircle (and bottom semicircle) broken up into 11 equally-spaced wedges.

     
     

     
     
105. 
     

     Question

     prevnext

     An angle measure, \(\theta\), equals \(\frac{-12\pi}{5}\) radians. A second angle measure, \(\phi\), is coterminal to \(\theta\), and \(0\le\phi<2\pi\). Which graph shows \(\phi\) in standard position?

     
     

     
     
     1. A
     2. B
     3. C
     4. D
     5. E
     6. F
     7. G
     8. H
     
     

     Solution

     Show solution

     The given angle measure is \(\theta=\frac{-12\pi}{5}\) radians. Because the given angle is less than 0, repeatedly add \(2\pi\) until getting a coterminal angle between 0 and \(2\pi\). It should be mentioned that \(2\pi = \frac{10\pi}{5}\), so we can add fractions with common denominators.

     \[\frac{-12\pi}{5}+\frac{10\pi}{5}=\frac{-2\pi}{5}\] \[\frac{-2\pi}{5}+\frac{10\pi}{5}=\frac{8\pi}{5}\]

     We found the coterminal angle \(\phi=\frac{8\pi}{5}\). I prefer to think of this as \(8\) times \(\frac{\pi}{5}\), so then I imagine the top semicircle (and bottom semicircle) broken up into 5 equally-spaced wedges.

     
     

     
     
106. 
     

     Question

     prevnext

     An angle measure, \(\theta\), equals \(\frac{76\pi}{11}\) radians. A second angle measure, \(\phi\), is coterminal to \(\theta\), and \(0\le\phi<2\pi\). Which graph shows \(\phi\) in standard position?

     
     

     
     
     1. A
     2. B
     3. C
     4. D
     5. E
     6. F
     7. G
     8. H
     
     

     Solution

     Show solution

     The given angle measure is \(\theta=\frac{76\pi}{11}\) radians. Because the given angle is more than \(2\pi\), repeatedly subtract \(2\pi\) until getting a coterminal angle between 0 and \(2\pi\). It should be mentioned that \(2\pi = \frac{22\pi}{11}\), so we can add fractions with common denominators.

     \[\frac{76\pi}{11}-\frac{22\pi}{11}=\frac{54\pi}{11}\] \[\frac{54\pi}{11}-\frac{22\pi}{11}=\frac{32\pi}{11}\] \[\frac{32\pi}{11}-\frac{22\pi}{11}=\frac{10\pi}{11}\]

     We found the coterminal angle \(\phi=\frac{10\pi}{11}\). I prefer to think of this as \(10\) times \(\frac{\pi}{11}\), so then I imagine the top semicircle (and bottom semicircle) broken up into 11 equally-spaced wedges.

     
     

     
     
107. 
     

     Question

     prevnext

     An angle measure, \(\theta\), equals \(\frac{28\pi}{9}\) radians. A second angle measure, \(\phi\), is coterminal to \(\theta\), and \(0\le\phi<2\pi\). Which graph shows \(\phi\) in standard position?

     
     

     
     
     1. A
     2. B
     3. C
     4. D
     5. E
     6. F
     7. G
     8. H
     
     

     Solution

     Show solution

     The given angle measure is \(\theta=\frac{28\pi}{9}\) radians. Because the given angle is more than \(2\pi\), repeatedly subtract \(2\pi\) until getting a coterminal angle between 0 and \(2\pi\). It should be mentioned that \(2\pi = \frac{18\pi}{9}\), so we can add fractions with common denominators.

     \[\frac{28\pi}{9}-\frac{18\pi}{9}=\frac{10\pi}{9}\]

     We found the coterminal angle \(\phi=\frac{10\pi}{9}\). I prefer to think of this as \(10\) times \(\frac{\pi}{9}\), so then I imagine the top semicircle (and bottom semicircle) broken up into 9 equally-spaced wedges.

     
     

     
     
108. 
     

     Question

     prevnext

     An angle measure, \(\theta\), equals \(\frac{-18\pi}{5}\) radians. A second angle measure, \(\phi\), is coterminal to \(\theta\), and \(0\le\phi<2\pi\). Which graph shows \(\phi\) in standard position?

     
     

     
     
     1. A
     2. B
     3. C
     4. D
     5. E
     6. F
     7. G
     8. H
     
     

     Solution

     Show solution

     The given angle measure is \(\theta=\frac{-18\pi}{5}\) radians. Because the given angle is less than 0, repeatedly add \(2\pi\) until getting a coterminal angle between 0 and \(2\pi\). It should be mentioned that \(2\pi = \frac{10\pi}{5}\), so we can add fractions with common denominators.

     \[\frac{-18\pi}{5}+\frac{10\pi}{5}=\frac{-8\pi}{5}\] \[\frac{-8\pi}{5}+\frac{10\pi}{5}=\frac{2\pi}{5}\]

     We found the coterminal angle \(\phi=\frac{2\pi}{5}\). I prefer to think of this as \(2\) times \(\frac{\pi}{5}\), so then I imagine the top semicircle (and bottom semicircle) broken up into 5 equally-spaced wedges.

     
     

     
     
109. 
     

     Question

     prevnext

     An angle measure, \(\theta\), equals \(\frac{-6\pi}{7}\) radians. A second angle measure, \(\phi\), is coterminal to \(\theta\), and \(0\le\phi<2\pi\). Which graph shows \(\phi\) in standard position?

     
     

     
     
     1. A
     2. B
     3. C
     4. D
     5. E
     6. F
     7. G
     8. H
     
     

     Solution

     Show solution

     The given angle measure is \(\theta=\frac{-6\pi}{7}\) radians. Because the given angle is less than 0, repeatedly add \(2\pi\) until getting a coterminal angle between 0 and \(2\pi\). It should be mentioned that \(2\pi = \frac{14\pi}{7}\), so we can add fractions with common denominators.

     \[\frac{-6\pi}{7}+\frac{14\pi}{7}=\frac{8\pi}{7}\]

     We found the coterminal angle \(\phi=\frac{8\pi}{7}\). I prefer to think of this as \(8\) times \(\frac{\pi}{7}\), so then I imagine the top semicircle (and bottom semicircle) broken up into 7 equally-spaced wedges.

     
     

     
     
110. 
     

     Question

     prevnext

     An angle measure, \(\theta\), equals \(\frac{-4\pi}{7}\) radians. A second angle measure, \(\phi\), is coterminal to \(\theta\), and \(0\le\phi<2\pi\). Which graph shows \(\phi\) in standard position?

     
     

     
     
     1. A
     2. B
     3. C
     4. D
     5. E
     6. F
     7. G
     8. H
     
     

     Solution

     Show solution

     The given angle measure is \(\theta=\frac{-4\pi}{7}\) radians. Because the given angle is less than 0, repeatedly add \(2\pi\) until getting a coterminal angle between 0 and \(2\pi\). It should be mentioned that \(2\pi = \frac{14\pi}{7}\), so we can add fractions with common denominators.

     \[\frac{-4\pi}{7}+\frac{14\pi}{7}=\frac{10\pi}{7}\]

     We found the coterminal angle \(\phi=\frac{10\pi}{7}\). I prefer to think of this as \(10\) times \(\frac{\pi}{7}\), so then I imagine the top semicircle (and bottom semicircle) broken up into 7 equally-spaced wedges.

     
     

     
     
111. 
     

     Question

     prevnext

     Which graph shows a \(\frac{-14\pi}{3}\) angle in standard position?

     
     

     
     
     1. A
     2. B
     3. C
     4. D
     5. E
     6. F
     7. G
     8. H
     
     

     Solution

     Show solution

     The given angle is \(\frac{-14\pi}{3}\). The angle is negative, so the angle rotates clockwise.

     Each wedge shown below corresponds to an angle of \(\frac{\pi}{3}\). So, we just need to turn across 14 wedges clockwise.

     
     

     
     
112. 
     

     Question

     prevnext

     Which graph shows a \(\frac{32\pi}{5}\) angle in standard position?

     
     

     
     
     1. A
     2. B
     3. C
     4. D
     5. E
     6. F
     7. G
     8. H
     
     

     Solution

     Show solution

     The given angle is \(\frac{32\pi}{5}\). The angle is positive, so the angle rotates counterclockwise.

     Each wedge shown below corresponds to an angle of \(\frac{\pi}{5}\). So, we just need to turn across 32 wedges counterclockwise.

     
     

     
     
113. 
     

     Question

     prevnext

     Which graph shows a \(\frac{24\pi}{7}\) angle in standard position?

     
     

     
     
     1. A
     2. B
     3. C
     4. D
     5. E
     6. F
     7. G
     8. H
     
     

     Solution

     Show solution

     The given angle is \(\frac{24\pi}{7}\). The angle is positive, so the angle rotates counterclockwise.

     Each wedge shown below corresponds to an angle of \(\frac{\pi}{7}\). So, we just need to turn across 24 wedges counterclockwise.

     
     

     
     
114. 
     

     Question

     prevnext

     Which graph shows a \(\frac{18\pi}{5}\) angle in standard position?

     
     

     
     
     1. A
     2. B
     3. C
     4. D
     5. E
     6. F
     7. G
     8. H
     
     

     Solution

     Show solution

     The given angle is \(\frac{18\pi}{5}\). The angle is positive, so the angle rotates counterclockwise.

     Each wedge shown below corresponds to an angle of \(\frac{\pi}{5}\). So, we just need to turn across 18 wedges counterclockwise.

     
     

     
     
115. 
     

     Question

     prevnext

     Which graph shows a \(\frac{17\pi}{7}\) angle in standard position?

     
     

     
     
     1. A
     2. B
     3. C
     4. D
     5. E
     6. F
     7. G
     8. H
     
     

     Solution

     Show solution

     The given angle is \(\frac{17\pi}{7}\). The angle is positive, so the angle rotates counterclockwise.

     Each wedge shown below corresponds to an angle of \(\frac{\pi}{7}\). So, we just need to turn across 17 wedges counterclockwise.

     
     

     
     
116. 
     

     Question

     prevnext

     Which graph shows a \(\frac{10\pi}{3}\) angle in standard position?

     
     

     
     
     1. A
     2. B
     3. C
     4. D
     5. E
     6. F
     7. G
     8. H
     
     

     Solution

     Show solution

     The given angle is \(\frac{10\pi}{3}\). The angle is positive, so the angle rotates counterclockwise.

     Each wedge shown below corresponds to an angle of \(\frac{\pi}{3}\). So, we just need to turn across 10 wedges counterclockwise.

     
     

     
     
117. 
     

     Question

     prevnext

     Which graph shows a \(\frac{19\pi}{6}\) angle in standard position?

     
     

     
     
     1. A
     2. B
     3. C
     4. D
     5. E
     6. F
     7. G
     8. H
     
     

     Solution

     Show solution

     The given angle is \(\frac{19\pi}{6}\). The angle is positive, so the angle rotates counterclockwise.

     Each wedge shown below corresponds to an angle of \(\frac{\pi}{6}\). So, we just need to turn across 19 wedges counterclockwise.

     
     

     
     
118. 
     

     Question

     prevnext

     Which graph shows a \(\frac{14\pi}{3}\) angle in standard position?

     
     

     
     
     1. A
     2. B
     3. C
     4. D
     5. E
     6. F
     7. G
     8. H
     
     

     Solution

     Show solution

     The given angle is \(\frac{14\pi}{3}\). The angle is positive, so the angle rotates counterclockwise.

     Each wedge shown below corresponds to an angle of \(\frac{\pi}{3}\). So, we just need to turn across 14 wedges counterclockwise.

     
     

     
     
119. 
     

     Question

     prevnext

     Which graph shows a \(\frac{52\pi}{7}\) angle in standard position?

     
     

     
     
     1. A
     2. B
     3. C
     4. D
     5. E
     6. F
     7. G
     8. H
     
     

     Solution

     Show solution

     The given angle is \(\frac{52\pi}{7}\). The angle is positive, so the angle rotates counterclockwise.

     Each wedge shown below corresponds to an angle of \(\frac{\pi}{7}\). So, we just need to turn across 52 wedges counterclockwise.

     
     

     
     
120. 
     

     Question

     prevnext

     Which graph shows a \(\frac{-11\pi}{6}\) angle in standard position?

     
     

     
     
     1. A
     2. B
     3. C
     4. D
     5. E
     6. F
     7. G
     8. H
     
     

     Solution

     Show solution

     The given angle is \(\frac{-11\pi}{6}\). The angle is negative, so the angle rotates clockwise.

     Each wedge shown below corresponds to an angle of \(\frac{\pi}{6}\). So, we just need to turn across 11 wedges clockwise.

     
     

     
     
121. 
     

     Question

     prevnext

     An arc is drawn along the circumference of a circle with a radius of 4 meters. The arc’s length is \(\frac{20\pi}{3}\) meters. What is the arc’s angle measure in radians?

     
     

     The angle measure, \(x\), can be expressed as \(\frac{a\pi}{b}\), where \(a\) and \(b\) are both whole numbers with no shared factors other than 1. Find \(a\) and \(b\).

     • \(a=\) -------- 1234567891011121314151617181920
     • \(b=\) -------- 1234567891011121314151617181920
     
     
     
     

     Solution

     Show solution

     Divide the arc length by the radius to get the angle measure (in radians).

     \[x = \frac{\frac{20\pi}{3}}{4}\]

     \[x = \frac{20\pi}{3} \cdot \frac{1}{4}\]

     \[x = \frac{20\pi}{3 \cdot 4}\]

     \[x = \frac{5\pi}{3}\]

     Thus, \(a = 5\) and \(b=3\).

     
     
122. 
     

     Question

     prevnext

     An arc is drawn along the circumference of a circle with a radius of 9 meters. The arc’s length is \(\frac{108\pi}{11}\) meters. What is the arc’s angle measure in radians?

     
     

     The angle measure, \(x\), can be expressed as \(\frac{a\pi}{b}\), where \(a\) and \(b\) are both whole numbers with no shared factors other than 1. Find \(a\) and \(b\).

     • \(a=\) -------- 1234567891011121314151617181920
     • \(b=\) -------- 1234567891011121314151617181920
     
     
     
     

     Solution

     Show solution

     Divide the arc length by the radius to get the angle measure (in radians).

     \[x = \frac{\frac{108\pi}{11}}{9}\]

     \[x = \frac{108\pi}{11} \cdot \frac{1}{9}\]

     \[x = \frac{108\pi}{11 \cdot 9}\]

     \[x = \frac{12\pi}{11}\]

     Thus, \(a = 12\) and \(b=11\).

     
     
123. 
     

     Question

     prevnext

     An arc is drawn along the circumference of a circle with a radius of 5 meters. The arc’s length is \(\frac{25\pi}{6}\) meters. What is the arc’s angle measure in radians?

     
     

     The angle measure, \(x\), can be expressed as \(\frac{a\pi}{b}\), where \(a\) and \(b\) are both whole numbers with no shared factors other than 1. Find \(a\) and \(b\).

     • \(a=\) -------- 1234567891011121314151617181920
     • \(b=\) -------- 1234567891011121314151617181920
     
     
     
     

     Solution

     Show solution

     Divide the arc length by the radius to get the angle measure (in radians).

     \[x = \frac{\frac{25\pi}{6}}{5}\]

     \[x = \frac{25\pi}{6} \cdot \frac{1}{5}\]

     \[x = \frac{25\pi}{6 \cdot 5}\]

     \[x = \frac{5\pi}{6}\]

     Thus, \(a = 5\) and \(b=6\).

     
     
124. 
     

     Question

     prevnext

     An arc is drawn along the circumference of a circle with a radius of 9 meters. The arc’s length is \(4\pi\) meters. What is the arc’s angle measure in radians?

     
     

     The angle measure, \(x\), can be expressed as \(\frac{a\pi}{b}\), where \(a\) and \(b\) are both whole numbers with no shared factors other than 1. Find \(a\) and \(b\).

     • \(a=\) -------- 1234567891011121314151617181920
     • \(b=\) -------- 1234567891011121314151617181920
     
     
     
     

     Solution

     Show solution

     Divide the arc length by the radius to get the angle measure (in radians).

     \[x = \frac{4\pi}{9}\]

     \[x = \frac{4\pi}{1} \cdot \frac{1}{9}\]

     \[x = \frac{4\pi}{1 \cdot 9}\]

     \[x = \frac{4\pi}{9}\]

     Thus, \(a = 4\) and \(b=9\).

     
     
125. 
     

     Question

     prevnext

     An arc is drawn along the circumference of a circle with a radius of 9 meters. The arc’s length is \(\frac{27\pi}{5}\) meters. What is the arc’s angle measure in radians?

     
     

     The angle measure, \(x\), can be expressed as \(\frac{a\pi}{b}\), where \(a\) and \(b\) are both whole numbers with no shared factors other than 1. Find \(a\) and \(b\).

     • \(a=\) -------- 1234567891011121314151617181920
     • \(b=\) -------- 1234567891011121314151617181920
     
     
     
     

     Solution

     Show solution

     Divide the arc length by the radius to get the angle measure (in radians).

     \[x = \frac{\frac{27\pi}{5}}{9}\]

     \[x = \frac{27\pi}{5} \cdot \frac{1}{9}\]

     \[x = \frac{27\pi}{5 \cdot 9}\]

     \[x = \frac{3\pi}{5}\]

     Thus, \(a = 3\) and \(b=5\).

     
     
126. 
     

     Question

     prevnext

     An arc is drawn along the circumference of a circle with a radius of 5 meters. The arc’s length is \(\frac{30\pi}{7}\) meters. What is the arc’s angle measure in radians?

     
     

     The angle measure, \(x\), can be expressed as \(\frac{a\pi}{b}\), where \(a\) and \(b\) are both whole numbers with no shared factors other than 1. Find \(a\) and \(b\).

     • \(a=\) -------- 1234567891011121314151617181920
     • \(b=\) -------- 1234567891011121314151617181920
     
     
     
     

     Solution

     Show solution

     Divide the arc length by the radius to get the angle measure (in radians).

     \[x = \frac{\frac{30\pi}{7}}{5}\]

     \[x = \frac{30\pi}{7} \cdot \frac{1}{5}\]

     \[x = \frac{30\pi}{7 \cdot 5}\]

     \[x = \frac{6\pi}{7}\]

     Thus, \(a = 6\) and \(b=7\).

     
     
127. 
     

     Question

     prevnext

     An arc is drawn along the circumference of a circle with a radius of 8 meters. The arc’s length is \(\frac{64\pi}{7}\) meters. What is the arc’s angle measure in radians?

     
     

     The angle measure, \(x\), can be expressed as \(\frac{a\pi}{b}\), where \(a\) and \(b\) are both whole numbers with no shared factors other than 1. Find \(a\) and \(b\).

     • \(a=\) -------- 1234567891011121314151617181920
     • \(b=\) -------- 1234567891011121314151617181920
     
     
     
     

     Solution

     Show solution

     Divide the arc length by the radius to get the angle measure (in radians).

     \[x = \frac{\frac{64\pi}{7}}{8}\]

     \[x = \frac{64\pi}{7} \cdot \frac{1}{8}\]

     \[x = \frac{64\pi}{7 \cdot 8}\]

     \[x = \frac{8\pi}{7}\]

     Thus, \(a = 8\) and \(b=7\).

     
     
128. 
     

     Question

     prevnext

     An arc is drawn along the circumference of a circle with a radius of 4 meters. The arc’s length is \(\frac{8\pi}{5}\) meters. What is the arc’s angle measure in radians?

     
     

     The angle measure, \(x\), can be expressed as \(\frac{a\pi}{b}\), where \(a\) and \(b\) are both whole numbers with no shared factors other than 1. Find \(a\) and \(b\).

     • \(a=\) -------- 1234567891011121314151617181920
     • \(b=\) -------- 1234567891011121314151617181920
     
     
     
     

     Solution

     Show solution

     Divide the arc length by the radius to get the angle measure (in radians).

     \[x = \frac{\frac{8\pi}{5}}{4}\]

     \[x = \frac{8\pi}{5} \cdot \frac{1}{4}\]

     \[x = \frac{8\pi}{5 \cdot 4}\]

     \[x = \frac{2\pi}{5}\]

     Thus, \(a = 2\) and \(b=5\).

     
     
129. 
     

     Question

     prevnext

     An arc is drawn along the circumference of a circle with a radius of 5 meters. The arc’s length is \(\frac{55\pi}{6}\) meters. What is the arc’s angle measure in radians?

     
     

     The angle measure, \(x\), can be expressed as \(\frac{a\pi}{b}\), where \(a\) and \(b\) are both whole numbers with no shared factors other than 1. Find \(a\) and \(b\).

     • \(a=\) -------- 1234567891011121314151617181920
     • \(b=\) -------- 1234567891011121314151617181920
     
     
     
     

     Solution

     Show solution

     Divide the arc length by the radius to get the angle measure (in radians).

     \[x = \frac{\frac{55\pi}{6}}{5}\]

     \[x = \frac{55\pi}{6} \cdot \frac{1}{5}\]

     \[x = \frac{55\pi}{6 \cdot 5}\]

     \[x = \frac{11\pi}{6}\]

     Thus, \(a = 11\) and \(b=6\).

     
     
130. 
     

     Question

     prevnext

     An arc is drawn along the circumference of a circle with a radius of 6 meters. The arc’s length is \(\frac{32\pi}{3}\) meters. What is the arc’s angle measure in radians?

     
     

     The angle measure, \(x\), can be expressed as \(\frac{a\pi}{b}\), where \(a\) and \(b\) are both whole numbers with no shared factors other than 1. Find \(a\) and \(b\).

     • \(a=\) -------- 1234567891011121314151617181920
     • \(b=\) -------- 1234567891011121314151617181920
     
     
     
     

     Solution

     Show solution

     Divide the arc length by the radius to get the angle measure (in radians).

     \[x = \frac{\frac{32\pi}{3}}{6}\]

     \[x = \frac{32\pi}{3} \cdot \frac{1}{6}\]

     \[x = \frac{32\pi}{3 \cdot 6}\]

     \[x = \frac{16\pi}{9}\]

     Thus, \(a = 16\) and \(b=9\).

     
     
131. 
     

     Question

     prevnext

     An arc is drawn along the circumference of a circle with a radius of \(r=6\) meters. The arc’s length is \(L=23.22\) meters. What is the corresponding sector’s area (labelled \(A\)) in square meters?

     
     

     Please give an exact answer (in \(\mathrm{m^2}\)).

     
     

     Solution

     Show solution

     Divide the arc length by the radius to get the angle measure (in radians).

     \[\theta = \frac{23.22}{6}\]

     \[\theta = 3.87\]

     Find the area of the entire circle. Then multiply that area by the fraction of the circle.

     \[A = \pi r^2 \cdot \frac{\theta}{2\pi}\]

     \[A = \pi (6)^2 \cdot \frac{3.87}{2\pi}\]

     \[A = 69.66\]

     If you are feeling clever, you might derive a simpler formula.

     \[\theta = \frac{L}{r}\] \[A ~=~ \pi r^2 \cdot \frac{L}{2\pi r} ~=~ \frac{rL}{2}\]

     So, the area of the sector is 69.66 \(\mathrm{m^2}\).

     
     
132. 
     

     Question

     prevnext

     An arc is drawn along the circumference of a circle with a radius of \(r=8\) meters. The arc’s length is \(L=28.16\) meters. What is the corresponding sector’s area (labelled \(A\)) in square meters?

     
     

     Please give an exact answer (in \(\mathrm{m^2}\)).

     
     

     Solution

     Show solution

     Divide the arc length by the radius to get the angle measure (in radians).

     \[\theta = \frac{28.16}{8}\]

     \[\theta = 3.52\]

     Find the area of the entire circle. Then multiply that area by the fraction of the circle.

     \[A = \pi r^2 \cdot \frac{\theta}{2\pi}\]

     \[A = \pi (8)^2 \cdot \frac{3.52}{2\pi}\]

     \[A = 112.64\]

     If you are feeling clever, you might derive a simpler formula.

     \[\theta = \frac{L}{r}\] \[A ~=~ \pi r^2 \cdot \frac{L}{2\pi r} ~=~ \frac{rL}{2}\]

     So, the area of the sector is 112.64 \(\mathrm{m^2}\).

     
     
133. 
     

     Question

     prevnext

     An arc is drawn along the circumference of a circle with a radius of \(r=6\) meters. The arc’s length is \(L=11.28\) meters. What is the corresponding sector’s area (labelled \(A\)) in square meters?

     
     

     Please give an exact answer (in \(\mathrm{m^2}\)).

     
     

     Solution

     Show solution

     Divide the arc length by the radius to get the angle measure (in radians).

     \[\theta = \frac{11.28}{6}\]

     \[\theta = 1.88\]

     Find the area of the entire circle. Then multiply that area by the fraction of the circle.

     \[A = \pi r^2 \cdot \frac{\theta}{2\pi}\]

     \[A = \pi (6)^2 \cdot \frac{1.88}{2\pi}\]

     \[A = 33.84\]

     If you are feeling clever, you might derive a simpler formula.

     \[\theta = \frac{L}{r}\] \[A ~=~ \pi r^2 \cdot \frac{L}{2\pi r} ~=~ \frac{rL}{2}\]

     So, the area of the sector is 33.84 \(\mathrm{m^2}\).

     
     
134. 
     

     Question

     prevnext

     An arc is drawn along the circumference of a circle with a radius of \(r=5\) meters. The arc’s length is \(L=5.8\) meters. What is the corresponding sector’s area (labelled \(A\)) in square meters?

     
     

     Please give an exact answer (in \(\mathrm{m^2}\)).

     
     

     Solution

     Show solution

     Divide the arc length by the radius to get the angle measure (in radians).

     \[\theta = \frac{5.8}{5}\]

     \[\theta = 1.16\]

     Find the area of the entire circle. Then multiply that area by the fraction of the circle.

     \[A = \pi r^2 \cdot \frac{\theta}{2\pi}\]

     \[A = \pi (5)^2 \cdot \frac{1.16}{2\pi}\]

     \[A = 14.5\]

     If you are feeling clever, you might derive a simpler formula.

     \[\theta = \frac{L}{r}\] \[A ~=~ \pi r^2 \cdot \frac{L}{2\pi r} ~=~ \frac{rL}{2}\]

     So, the area of the sector is 14.5 \(\mathrm{m^2}\).

     
     
135. 
     

     Question

     prevnext

     An arc is drawn along the circumference of a circle with a radius of \(r=5\) meters. The arc’s length is \(L=5.1\) meters. What is the corresponding sector’s area (labelled \(A\)) in square meters?

     
     

     Please give an exact answer (in \(\mathrm{m^2}\)).

     
     

     Solution

     Show solution

     Divide the arc length by the radius to get the angle measure (in radians).

     \[\theta = \frac{5.1}{5}\]

     \[\theta = 1.02\]

     Find the area of the entire circle. Then multiply that area by the fraction of the circle.

     \[A = \pi r^2 \cdot \frac{\theta}{2\pi}\]

     \[A = \pi (5)^2 \cdot \frac{1.02}{2\pi}\]

     \[A = 12.75\]

     If you are feeling clever, you might derive a simpler formula.

     \[\theta = \frac{L}{r}\] \[A ~=~ \pi r^2 \cdot \frac{L}{2\pi r} ~=~ \frac{rL}{2}\]

     So, the area of the sector is 12.75 \(\mathrm{m^2}\).

     
     
136. 
     

     Question

     prevnext

     An arc is drawn along the circumference of a circle with a radius of \(r=4\) meters. The arc’s length is \(L=5.28\) meters. What is the corresponding sector’s area (labelled \(A\)) in square meters?

     
     

     Please give an exact answer (in \(\mathrm{m^2}\)).

     
     

     Solution

     Show solution

     Divide the arc length by the radius to get the angle measure (in radians).

     \[\theta = \frac{5.28}{4}\]

     \[\theta = 1.32\]

     Find the area of the entire circle. Then multiply that area by the fraction of the circle.

     \[A = \pi r^2 \cdot \frac{\theta}{2\pi}\]

     \[A = \pi (4)^2 \cdot \frac{1.32}{2\pi}\]

     \[A = 10.56\]

     If you are feeling clever, you might derive a simpler formula.

     \[\theta = \frac{L}{r}\] \[A ~=~ \pi r^2 \cdot \frac{L}{2\pi r} ~=~ \frac{rL}{2}\]

     So, the area of the sector is 10.56 \(\mathrm{m^2}\).

     
     
137. 
     

     Question

     prevnext

     An arc is drawn along the circumference of a circle with a radius of \(r=6\) meters. The arc’s length is \(L=7.86\) meters. What is the corresponding sector’s area (labelled \(A\)) in square meters?

     
     

     Please give an exact answer (in \(\mathrm{m^2}\)).

     
     

     Solution

     Show solution

     Divide the arc length by the radius to get the angle measure (in radians).

     \[\theta = \frac{7.86}{6}\]

     \[\theta = 1.31\]

     Find the area of the entire circle. Then multiply that area by the fraction of the circle.

     \[A = \pi r^2 \cdot \frac{\theta}{2\pi}\]

     \[A = \pi (6)^2 \cdot \frac{1.31}{2\pi}\]

     \[A = 23.58\]

     If you are feeling clever, you might derive a simpler formula.

     \[\theta = \frac{L}{r}\] \[A ~=~ \pi r^2 \cdot \frac{L}{2\pi r} ~=~ \frac{rL}{2}\]

     So, the area of the sector is 23.58 \(\mathrm{m^2}\).

     
     
138. 
     

     Question

     prevnext

     An arc is drawn along the circumference of a circle with a radius of \(r=6\) meters. The arc’s length is \(L=30.6\) meters. What is the corresponding sector’s area (labelled \(A\)) in square meters?

     
     

     Please give an exact answer (in \(\mathrm{m^2}\)).

     
     

     Solution

     Show solution

     Divide the arc length by the radius to get the angle measure (in radians).

     \[\theta = \frac{30.6}{6}\]

     \[\theta = 5.1\]

     Find the area of the entire circle. Then multiply that area by the fraction of the circle.

     \[A = \pi r^2 \cdot \frac{\theta}{2\pi}\]

     \[A = \pi (6)^2 \cdot \frac{5.1}{2\pi}\]

     \[A = 91.8\]

     If you are feeling clever, you might derive a simpler formula.

     \[\theta = \frac{L}{r}\] \[A ~=~ \pi r^2 \cdot \frac{L}{2\pi r} ~=~ \frac{rL}{2}\]

     So, the area of the sector is 91.8 \(\mathrm{m^2}\).

     
     
139. 
     

     Question

     prevnext

     An arc is drawn along the circumference of a circle with a radius of \(r=8\) meters. The arc’s length is \(L=18.8\) meters. What is the corresponding sector’s area (labelled \(A\)) in square meters?

     
     

     Please give an exact answer (in \(\mathrm{m^2}\)).

     
     

     Solution

     Show solution

     Divide the arc length by the radius to get the angle measure (in radians).

     \[\theta = \frac{18.8}{8}\]

     \[\theta = 2.35\]

     Find the area of the entire circle. Then multiply that area by the fraction of the circle.

     \[A = \pi r^2 \cdot \frac{\theta}{2\pi}\]

     \[A = \pi (8)^2 \cdot \frac{2.35}{2\pi}\]

     \[A = 75.2\]

     If you are feeling clever, you might derive a simpler formula.

     \[\theta = \frac{L}{r}\] \[A ~=~ \pi r^2 \cdot \frac{L}{2\pi r} ~=~ \frac{rL}{2}\]

     So, the area of the sector is 75.2 \(\mathrm{m^2}\).

     
     
140. 
     

     Question

     prevnext

     An arc is drawn along the circumference of a circle with a radius of \(r=6\) meters. The arc’s length is \(L=31.2\) meters. What is the corresponding sector’s area (labelled \(A\)) in square meters?

     
     

     Please give an exact answer (in \(\mathrm{m^2}\)).

     
     

     Solution

     Show solution

     Divide the arc length by the radius to get the angle measure (in radians).

     \[\theta = \frac{31.2}{6}\]

     \[\theta = 5.2\]

     Find the area of the entire circle. Then multiply that area by the fraction of the circle.

     \[A = \pi r^2 \cdot \frac{\theta}{2\pi}\]

     \[A = \pi (6)^2 \cdot \frac{5.2}{2\pi}\]

     \[A = 93.6\]

     If you are feeling clever, you might derive a simpler formula.

     \[\theta = \frac{L}{r}\] \[A ~=~ \pi r^2 \cdot \frac{L}{2\pi r} ~=~ \frac{rL}{2}\]

     So, the area of the sector is 93.6 \(\mathrm{m^2}\).

     
     
141. 
     

     Question

     prevnext

     A circle with radius \(R=4\) ft is cut to a sector with an angle of \(\theta=\frac{4\pi}{3}\) radians.

     
     

     By connecting the initial side and terminal side, the sector is bent (in 3 dimensions) into a right-circular cone with height \(h\) and radius \(r\). Notice the cone’s slant height equals the radius of the original circle (\(R\)) and the circumference of the cone matches the arc length (\(L\)) of the original circle.

     
     

     Find the ratio of the cone’s height to the cone’s radius.

     \[\frac{h}{r} ~=~ ?\]

     The tolerance is \(\pm 0.01\).

     
     

     Solution

     Show solution

     First, find the arc length.

     \[L ~=~ \frac{4\pi}{3} \cdot 4 ~=~ \frac{16\pi}{3}\]

     We know that a circle’s circumference is twice its radius times \(\pi\). Look at the circular base of the cone, and use the circumference formula to find its radius. Remember \(L\) is the circumference of the cone’s base.

     \[L=2\pi r\]

     Divide both sides by \(2\pi\) and switch sides to get \(r=\frac{L}{2\pi}\).

     \[r ~=~ \frac{L}{2\pi} ~=~ \frac{\frac{16\pi}{3}}{2\pi}\]

     \[r ~=~ \frac{8}{3}\]

     To get the height of the cone, use Pythagorean theorem, which states a right triangle with legs \(a\) and \(b\) and hypotenuse \(c\) is constrained by \(a^2+b^2=c^2\).

     \[\left(\frac{8}{3}\right)^2 + h^2 = (4)^2\]

     Solve for \(h\).

     \[h = \sqrt{(4)^2-\left(\frac{8}{3}\right)^2}\]

     \[h\approx 2.981424\]

     Find the ratio.

     \[\frac{h}{r} ~=~ \frac{\sqrt{(4)^2-\left(\frac{8}{3}\right)^2}}{\frac{8}{3}}\]

     \[\frac{h}{r} \approx 1.118034\]

     Fancy stuff

     With some thought, you might convince yourself that this ratio \(\frac{h}{r}\) should be independent of the radius of the original circle. For example, if you measured the radius \(R\) in inches (instead of feet), the \(\frac{h}{r}\) ratio shouldn’t change. Or if you doubled \(R\), it seems \(\frac{h}{r}\) should be the same.

     By approaching this problem more algebraically, you might reach some interesting insights. For example, if we define \(f\) to be the fraction of the original circle.

     \[f = \frac{\theta}{2\pi}\]

     You can show that the ratio of radii also equals \(f\).

     \[\frac{r}{R} = f\]

     Substituting \(r=Rf\), you can generalize the formula to find \(h\) in terms of \(R\) and \(f\): \[h ~=~ \sqrt{R^2-r^2} ~=~ \sqrt{R^2-(fR)^2} ~=~ R\sqrt{1-f^2}\]

     Using that information, you can derive a simple formula for \(\frac{h}{r}\) that only depends on \(f\).

     
     
142. 
     

     Question

     prevnext

     A circle with radius \(R=7\) ft is cut to a sector with an angle of \(\theta=\frac{3\pi}{4}\) radians.

     
     

     By connecting the initial side and terminal side, the sector is bent (in 3 dimensions) into a right-circular cone with height \(h\) and radius \(r\). Notice the cone’s slant height equals the radius of the original circle (\(R\)) and the circumference of the cone matches the arc length (\(L\)) of the original circle.

     
     

     Find the ratio of the cone’s height to the cone’s radius.

     \[\frac{h}{r} ~=~ ?\]

     The tolerance is \(\pm 0.01\).

     
     

     Solution

     Show solution

     First, find the arc length.

     \[L ~=~ \frac{3\pi}{4} \cdot 7 ~=~ \frac{21\pi}{4}\]

     We know that a circle’s circumference is twice its radius times \(\pi\). Look at the circular base of the cone, and use the circumference formula to find its radius. Remember \(L\) is the circumference of the cone’s base.

     \[L=2\pi r\]

     Divide both sides by \(2\pi\) and switch sides to get \(r=\frac{L}{2\pi}\).

     \[r ~=~ \frac{L}{2\pi} ~=~ \frac{\frac{21\pi}{4}}{2\pi}\]

     \[r ~=~ \frac{21}{8}\]

     To get the height of the cone, use Pythagorean theorem, which states a right triangle with legs \(a\) and \(b\) and hypotenuse \(c\) is constrained by \(a^2+b^2=c^2\).

     \[\left(\frac{21}{8}\right)^2 + h^2 = (7)^2\]

     Solve for \(h\).

     \[h = \sqrt{(7)^2-\left(\frac{21}{8}\right)^2}\]

     \[h\approx 6.4891737\]

     Find the ratio.

     \[\frac{h}{r} ~=~ \frac{\sqrt{(7)^2-\left(\frac{21}{8}\right)^2}}{\frac{21}{8}}\]

     \[\frac{h}{r} \approx 2.4720662\]

     Fancy stuff

     With some thought, you might convince yourself that this ratio \(\frac{h}{r}\) should be independent of the radius of the original circle. For example, if you measured the radius \(R\) in inches (instead of feet), the \(\frac{h}{r}\) ratio shouldn’t change. Or if you doubled \(R\), it seems \(\frac{h}{r}\) should be the same.

     By approaching this problem more algebraically, you might reach some interesting insights. For example, if we define \(f\) to be the fraction of the original circle.

     \[f = \frac{\theta}{2\pi}\]

     You can show that the ratio of radii also equals \(f\).

     \[\frac{r}{R} = f\]

     Substituting \(r=Rf\), you can generalize the formula to find \(h\) in terms of \(R\) and \(f\): \[h ~=~ \sqrt{R^2-r^2} ~=~ \sqrt{R^2-(fR)^2} ~=~ R\sqrt{1-f^2}\]

     Using that information, you can derive a simple formula for \(\frac{h}{r}\) that only depends on \(f\).

     
     
143. 
     

     Question

     prevnext

     A circle with radius \(R=8\) ft is cut to a sector with an angle of \(\theta=\frac{11\pi}{9}\) radians.

     
     

     By connecting the initial side and terminal side, the sector is bent (in 3 dimensions) into a right-circular cone with height \(h\) and radius \(r\). Notice the cone’s slant height equals the radius of the original circle (\(R\)) and the circumference of the cone matches the arc length (\(L\)) of the original circle.

     
     

     Find the ratio of the cone’s height to the cone’s radius.

     \[\frac{h}{r} ~=~ ?\]

     The tolerance is \(\pm 0.01\).

     
     

     Solution

     Show solution

     First, find the arc length.

     \[L ~=~ \frac{11\pi}{9} \cdot 8 ~=~ \frac{88\pi}{9}\]

     We know that a circle’s circumference is twice its radius times \(\pi\). Look at the circular base of the cone, and use the circumference formula to find its radius. Remember \(L\) is the circumference of the cone’s base.

     \[L=2\pi r\]

     Divide both sides by \(2\pi\) and switch sides to get \(r=\frac{L}{2\pi}\).

     \[r ~=~ \frac{L}{2\pi} ~=~ \frac{\frac{88\pi}{9}}{2\pi}\]

     \[r ~=~ \frac{44}{9}\]

     To get the height of the cone, use Pythagorean theorem, which states a right triangle with legs \(a\) and \(b\) and hypotenuse \(c\) is constrained by \(a^2+b^2=c^2\).

     \[\left(\frac{44}{9}\right)^2 + h^2 = (8)^2\]

     Solve for \(h\).

     \[h = \sqrt{(8)^2-\left(\frac{44}{9}\right)^2}\]

     \[h\approx 6.3323586\]

     Find the ratio.

     \[\frac{h}{r} ~=~ \frac{\sqrt{(8)^2-\left(\frac{44}{9}\right)^2}}{\frac{44}{9}}\]

     \[\frac{h}{r} \approx 1.2952552\]

     Fancy stuff

     With some thought, you might convince yourself that this ratio \(\frac{h}{r}\) should be independent of the radius of the original circle. For example, if you measured the radius \(R\) in inches (instead of feet), the \(\frac{h}{r}\) ratio shouldn’t change. Or if you doubled \(R\), it seems \(\frac{h}{r}\) should be the same.

     By approaching this problem more algebraically, you might reach some interesting insights. For example, if we define \(f\) to be the fraction of the original circle.

     \[f = \frac{\theta}{2\pi}\]

     You can show that the ratio of radii also equals \(f\).

     \[\frac{r}{R} = f\]

     Substituting \(r=Rf\), you can generalize the formula to find \(h\) in terms of \(R\) and \(f\): \[h ~=~ \sqrt{R^2-r^2} ~=~ \sqrt{R^2-(fR)^2} ~=~ R\sqrt{1-f^2}\]

     Using that information, you can derive a simple formula for \(\frac{h}{r}\) that only depends on \(f\).

     
     
144. 
     

     Question

     prevnext

     A circle with radius \(R=6\) ft is cut to a sector with an angle of \(\theta=\frac{8\pi}{7}\) radians.

     
     

     By connecting the initial side and terminal side, the sector is bent (in 3 dimensions) into a right-circular cone with height \(h\) and radius \(r\). Notice the cone’s slant height equals the radius of the original circle (\(R\)) and the circumference of the cone matches the arc length (\(L\)) of the original circle.

     
     

     Find the ratio of the cone’s height to the cone’s radius.

     \[\frac{h}{r} ~=~ ?\]

     The tolerance is \(\pm 0.01\).

     
     

     Solution

     Show solution

     First, find the arc length.

     \[L ~=~ \frac{8\pi}{7} \cdot 6 ~=~ \frac{48\pi}{7}\]

     We know that a circle’s circumference is twice its radius times \(\pi\). Look at the circular base of the cone, and use the circumference formula to find its radius. Remember \(L\) is the circumference of the cone’s base.

     \[L=2\pi r\]

     Divide both sides by \(2\pi\) and switch sides to get \(r=\frac{L}{2\pi}\).

     \[r ~=~ \frac{L}{2\pi} ~=~ \frac{\frac{48\pi}{7}}{2\pi}\]

     \[r ~=~ \frac{24}{7}\]

     To get the height of the cone, use Pythagorean theorem, which states a right triangle with legs \(a\) and \(b\) and hypotenuse \(c\) is constrained by \(a^2+b^2=c^2\).

     \[\left(\frac{24}{7}\right)^2 + h^2 = (6)^2\]

     Solve for \(h\).

     \[h = \sqrt{(6)^2-\left(\frac{24}{7}\right)^2}\]

     \[h\approx 4.9239108\]

     Find the ratio.

     \[\frac{h}{r} ~=~ \frac{\sqrt{(6)^2-\left(\frac{24}{7}\right)^2}}{\frac{24}{7}}\]

     \[\frac{h}{r} \approx 1.4361407\]

     Fancy stuff

     With some thought, you might convince yourself that this ratio \(\frac{h}{r}\) should be independent of the radius of the original circle. For example, if you measured the radius \(R\) in inches (instead of feet), the \(\frac{h}{r}\) ratio shouldn’t change. Or if you doubled \(R\), it seems \(\frac{h}{r}\) should be the same.

     By approaching this problem more algebraically, you might reach some interesting insights. For example, if we define \(f\) to be the fraction of the original circle.

     \[f = \frac{\theta}{2\pi}\]

     You can show that the ratio of radii also equals \(f\).

     \[\frac{r}{R} = f\]

     Substituting \(r=Rf\), you can generalize the formula to find \(h\) in terms of \(R\) and \(f\): \[h ~=~ \sqrt{R^2-r^2} ~=~ \sqrt{R^2-(fR)^2} ~=~ R\sqrt{1-f^2}\]

     Using that information, you can derive a simple formula for \(\frac{h}{r}\) that only depends on \(f\).

     
     
145. 
     

     Question

     prevnext

     A circle with radius \(R=9\) ft is cut to a sector with an angle of \(\theta=\frac{11\pi}{9}\) radians.

     
     

     By connecting the initial side and terminal side, the sector is bent (in 3 dimensions) into a right-circular cone with height \(h\) and radius \(r\). Notice the cone’s slant height equals the radius of the original circle (\(R\)) and the circumference of the cone matches the arc length (\(L\)) of the original circle.

     
     

     Find the ratio of the cone’s height to the cone’s radius.

     \[\frac{h}{r} ~=~ ?\]

     The tolerance is \(\pm 0.01\).

     
     

     Solution

     Show solution

     First, find the arc length.

     \[L ~=~ \frac{11\pi}{9} \cdot 9 ~=~ 11\pi\]

     We know that a circle’s circumference is twice its radius times \(\pi\). Look at the circular base of the cone, and use the circumference formula to find its radius. Remember \(L\) is the circumference of the cone’s base.

     \[L=2\pi r\]

     Divide both sides by \(2\pi\) and switch sides to get \(r=\frac{L}{2\pi}\).

     \[r ~=~ \frac{L}{2\pi} ~=~ \frac{11\pi}{2\pi}\]

     \[r ~=~ \frac{11}{2}\]

     To get the height of the cone, use Pythagorean theorem, which states a right triangle with legs \(a\) and \(b\) and hypotenuse \(c\) is constrained by \(a^2+b^2=c^2\).

     \[\left(\frac{11}{2}\right)^2 + h^2 = (9)^2\]

     Solve for \(h\).

     \[h = \sqrt{(9)^2-\left(\frac{11}{2}\right)^2}\]

     \[h\approx 7.1239034\]

     Find the ratio.

     \[\frac{h}{r} ~=~ \frac{\sqrt{(9)^2-\left(\frac{11}{2}\right)^2}}{\frac{11}{2}}\]

     \[\frac{h}{r} \approx 1.2952552\]

     Fancy stuff

     With some thought, you might convince yourself that this ratio \(\frac{h}{r}\) should be independent of the radius of the original circle. For example, if you measured the radius \(R\) in inches (instead of feet), the \(\frac{h}{r}\) ratio shouldn’t change. Or if you doubled \(R\), it seems \(\frac{h}{r}\) should be the same.

     By approaching this problem more algebraically, you might reach some interesting insights. For example, if we define \(f\) to be the fraction of the original circle.

     \[f = \frac{\theta}{2\pi}\]

     You can show that the ratio of radii also equals \(f\).

     \[\frac{r}{R} = f\]

     Substituting \(r=Rf\), you can generalize the formula to find \(h\) in terms of \(R\) and \(f\): \[h ~=~ \sqrt{R^2-r^2} ~=~ \sqrt{R^2-(fR)^2} ~=~ R\sqrt{1-f^2}\]

     Using that information, you can derive a simple formula for \(\frac{h}{r}\) that only depends on \(f\).

     
     
146. 
     

     Question

     prevnext

     A circle with radius \(R=8\) ft is cut to a sector with an angle of \(\theta=\frac{15\pi}{8}\) radians.

     
     

     By connecting the initial side and terminal side, the sector is bent (in 3 dimensions) into a right-circular cone with height \(h\) and radius \(r\). Notice the cone’s slant height equals the radius of the original circle (\(R\)) and the circumference of the cone matches the arc length (\(L\)) of the original circle.

     
     

     Find the ratio of the cone’s height to the cone’s radius.

     \[\frac{h}{r} ~=~ ?\]

     The tolerance is \(\pm 0.01\).

     
     

     Solution

     Show solution

     First, find the arc length.

     \[L ~=~ \frac{15\pi}{8} \cdot 8 ~=~ 15\pi\]

     We know that a circle’s circumference is twice its radius times \(\pi\). Look at the circular base of the cone, and use the circumference formula to find its radius. Remember \(L\) is the circumference of the cone’s base.

     \[L=2\pi r\]

     Divide both sides by \(2\pi\) and switch sides to get \(r=\frac{L}{2\pi}\).

     \[r ~=~ \frac{L}{2\pi} ~=~ \frac{15\pi}{2\pi}\]

     \[r ~=~ \frac{15}{2}\]

     To get the height of the cone, use Pythagorean theorem, which states a right triangle with legs \(a\) and \(b\) and hypotenuse \(c\) is constrained by \(a^2+b^2=c^2\).

     \[\left(\frac{15}{2}\right)^2 + h^2 = (8)^2\]

     Solve for \(h\).

     \[h = \sqrt{(8)^2-\left(\frac{15}{2}\right)^2}\]

     \[h\approx 2.7838822\]

     Find the ratio.

     \[\frac{h}{r} ~=~ \frac{\sqrt{(8)^2-\left(\frac{15}{2}\right)^2}}{\frac{15}{2}}\]

     \[\frac{h}{r} \approx 0.3711843\]

     Fancy stuff

     With some thought, you might convince yourself that this ratio \(\frac{h}{r}\) should be independent of the radius of the original circle. For example, if you measured the radius \(R\) in inches (instead of feet), the \(\frac{h}{r}\) ratio shouldn’t change. Or if you doubled \(R\), it seems \(\frac{h}{r}\) should be the same.

     By approaching this problem more algebraically, you might reach some interesting insights. For example, if we define \(f\) to be the fraction of the original circle.

     \[f = \frac{\theta}{2\pi}\]

     You can show that the ratio of radii also equals \(f\).

     \[\frac{r}{R} = f\]

     Substituting \(r=Rf\), you can generalize the formula to find \(h\) in terms of \(R\) and \(f\): \[h ~=~ \sqrt{R^2-r^2} ~=~ \sqrt{R^2-(fR)^2} ~=~ R\sqrt{1-f^2}\]

     Using that information, you can derive a simple formula for \(\frac{h}{r}\) that only depends on \(f\).

     
     
147. 
     

     Question

     prevnext

     A circle with radius \(R=6\) ft is cut to a sector with an angle of \(\theta=\frac{15\pi}{8}\) radians.

     
     

     By connecting the initial side and terminal side, the sector is bent (in 3 dimensions) into a right-circular cone with height \(h\) and radius \(r\). Notice the cone’s slant height equals the radius of the original circle (\(R\)) and the circumference of the cone matches the arc length (\(L\)) of the original circle.

     
     

     Find the ratio of the cone’s height to the cone’s radius.

     \[\frac{h}{r} ~=~ ?\]

     The tolerance is \(\pm 0.01\).

     
     

     Solution

     Show solution

     First, find the arc length.

     \[L ~=~ \frac{15\pi}{8} \cdot 6 ~=~ \frac{45\pi}{4}\]

     We know that a circle’s circumference is twice its radius times \(\pi\). Look at the circular base of the cone, and use the circumference formula to find its radius. Remember \(L\) is the circumference of the cone’s base.

     \[L=2\pi r\]

     Divide both sides by \(2\pi\) and switch sides to get \(r=\frac{L}{2\pi}\).

     \[r ~=~ \frac{L}{2\pi} ~=~ \frac{\frac{45\pi}{4}}{2\pi}\]

     \[r ~=~ \frac{45}{8}\]

     To get the height of the cone, use Pythagorean theorem, which states a right triangle with legs \(a\) and \(b\) and hypotenuse \(c\) is constrained by \(a^2+b^2=c^2\).

     \[\left(\frac{45}{8}\right)^2 + h^2 = (6)^2\]

     Solve for \(h\).

     \[h = \sqrt{(6)^2-\left(\frac{45}{8}\right)^2}\]

     \[h\approx 2.0879116\]

     Find the ratio.

     \[\frac{h}{r} ~=~ \frac{\sqrt{(6)^2-\left(\frac{45}{8}\right)^2}}{\frac{45}{8}}\]

     \[\frac{h}{r} \approx 0.3711843\]

     Fancy stuff

     With some thought, you might convince yourself that this ratio \(\frac{h}{r}\) should be independent of the radius of the original circle. For example, if you measured the radius \(R\) in inches (instead of feet), the \(\frac{h}{r}\) ratio shouldn’t change. Or if you doubled \(R\), it seems \(\frac{h}{r}\) should be the same.

     By approaching this problem more algebraically, you might reach some interesting insights. For example, if we define \(f\) to be the fraction of the original circle.

     \[f = \frac{\theta}{2\pi}\]

     You can show that the ratio of radii also equals \(f\).

     \[\frac{r}{R} = f\]

     Substituting \(r=Rf\), you can generalize the formula to find \(h\) in terms of \(R\) and \(f\): \[h ~=~ \sqrt{R^2-r^2} ~=~ \sqrt{R^2-(fR)^2} ~=~ R\sqrt{1-f^2}\]

     Using that information, you can derive a simple formula for \(\frac{h}{r}\) that only depends on \(f\).

     
     
148. 
     

     Question

     prevnext

     A circle with radius \(R=5\) ft is cut to a sector with an angle of \(\theta=\frac{3\pi}{5}\) radians.

     
     

     By connecting the initial side and terminal side, the sector is bent (in 3 dimensions) into a right-circular cone with height \(h\) and radius \(r\). Notice the cone’s slant height equals the radius of the original circle (\(R\)) and the circumference of the cone matches the arc length (\(L\)) of the original circle.

     
     

     Find the ratio of the cone’s height to the cone’s radius.

     \[\frac{h}{r} ~=~ ?\]

     The tolerance is \(\pm 0.01\).

     
     

     Solution

     Show solution

     First, find the arc length.

     \[L ~=~ \frac{3\pi}{5} \cdot 5 ~=~ 3\pi\]

     We know that a circle’s circumference is twice its radius times \(\pi\). Look at the circular base of the cone, and use the circumference formula to find its radius. Remember \(L\) is the circumference of the cone’s base.

     \[L=2\pi r\]

     Divide both sides by \(2\pi\) and switch sides to get \(r=\frac{L}{2\pi}\).

     \[r ~=~ \frac{L}{2\pi} ~=~ \frac{3\pi}{2\pi}\]

     \[r ~=~ \frac{3}{2}\]

     To get the height of the cone, use Pythagorean theorem, which states a right triangle with legs \(a\) and \(b\) and hypotenuse \(c\) is constrained by \(a^2+b^2=c^2\).

     \[\left(\frac{3}{2}\right)^2 + h^2 = (5)^2\]

     Solve for \(h\).

     \[h = \sqrt{(5)^2-\left(\frac{3}{2}\right)^2}\]

     \[h\approx 4.769696\]

     Find the ratio.

     \[\frac{h}{r} ~=~ \frac{\sqrt{(5)^2-\left(\frac{3}{2}\right)^2}}{\frac{3}{2}}\]

     \[\frac{h}{r} \approx 3.1797973\]

     Fancy stuff

     With some thought, you might convince yourself that this ratio \(\frac{h}{r}\) should be independent of the radius of the original circle. For example, if you measured the radius \(R\) in inches (instead of feet), the \(\frac{h}{r}\) ratio shouldn’t change. Or if you doubled \(R\), it seems \(\frac{h}{r}\) should be the same.

     By approaching this problem more algebraically, you might reach some interesting insights. For example, if we define \(f\) to be the fraction of the original circle.

     \[f = \frac{\theta}{2\pi}\]

     You can show that the ratio of radii also equals \(f\).

     \[\frac{r}{R} = f\]

     Substituting \(r=Rf\), you can generalize the formula to find \(h\) in terms of \(R\) and \(f\): \[h ~=~ \sqrt{R^2-r^2} ~=~ \sqrt{R^2-(fR)^2} ~=~ R\sqrt{1-f^2}\]

     Using that information, you can derive a simple formula for \(\frac{h}{r}\) that only depends on \(f\).

     
     
149. 
     

     Question

     prevnext

     A circle with radius \(R=4\) ft is cut to a sector with an angle of \(\theta=\frac{2\pi}{3}\) radians.

     
     

     By connecting the initial side and terminal side, the sector is bent (in 3 dimensions) into a right-circular cone with height \(h\) and radius \(r\). Notice the cone’s slant height equals the radius of the original circle (\(R\)) and the circumference of the cone matches the arc length (\(L\)) of the original circle.

     
     

     Find the ratio of the cone’s height to the cone’s radius.

     \[\frac{h}{r} ~=~ ?\]

     The tolerance is \(\pm 0.01\).

     
     

     Solution

     Show solution

     First, find the arc length.

     \[L ~=~ \frac{2\pi}{3} \cdot 4 ~=~ \frac{8\pi}{3}\]

     We know that a circle’s circumference is twice its radius times \(\pi\). Look at the circular base of the cone, and use the circumference formula to find its radius. Remember \(L\) is the circumference of the cone’s base.

     \[L=2\pi r\]

     Divide both sides by \(2\pi\) and switch sides to get \(r=\frac{L}{2\pi}\).

     \[r ~=~ \frac{L}{2\pi} ~=~ \frac{\frac{8\pi}{3}}{2\pi}\]

     \[r ~=~ \frac{4}{3}\]

     To get the height of the cone, use Pythagorean theorem, which states a right triangle with legs \(a\) and \(b\) and hypotenuse \(c\) is constrained by \(a^2+b^2=c^2\).

     \[\left(\frac{4}{3}\right)^2 + h^2 = (4)^2\]

     Solve for \(h\).

     \[h = \sqrt{(4)^2-\left(\frac{4}{3}\right)^2}\]

     \[h\approx 3.7712362\]

     Find the ratio.

     \[\frac{h}{r} ~=~ \frac{\sqrt{(4)^2-\left(\frac{4}{3}\right)^2}}{\frac{4}{3}}\]

     \[\frac{h}{r} \approx 2.8284271\]

     Fancy stuff

     With some thought, you might convince yourself that this ratio \(\frac{h}{r}\) should be independent of the radius of the original circle. For example, if you measured the radius \(R\) in inches (instead of feet), the \(\frac{h}{r}\) ratio shouldn’t change. Or if you doubled \(R\), it seems \(\frac{h}{r}\) should be the same.

     By approaching this problem more algebraically, you might reach some interesting insights. For example, if we define \(f\) to be the fraction of the original circle.

     \[f = \frac{\theta}{2\pi}\]

     You can show that the ratio of radii also equals \(f\).

     \[\frac{r}{R} = f\]

     Substituting \(r=Rf\), you can generalize the formula to find \(h\) in terms of \(R\) and \(f\): \[h ~=~ \sqrt{R^2-r^2} ~=~ \sqrt{R^2-(fR)^2} ~=~ R\sqrt{1-f^2}\]

     Using that information, you can derive a simple formula for \(\frac{h}{r}\) that only depends on \(f\).

     
     
150. 
     

     Question

     prevnext

     A circle with radius \(R=7\) ft is cut to a sector with an angle of \(\theta=\frac{7\pi}{8}\) radians.

     
     

     By connecting the initial side and terminal side, the sector is bent (in 3 dimensions) into a right-circular cone with height \(h\) and radius \(r\). Notice the cone’s slant height equals the radius of the original circle (\(R\)) and the circumference of the cone matches the arc length (\(L\)) of the original circle.

     
     

     Find the ratio of the cone’s height to the cone’s radius.

     \[\frac{h}{r} ~=~ ?\]

     The tolerance is \(\pm 0.01\).

     
     

     Solution

     Show solution

     First, find the arc length.

     \[L ~=~ \frac{7\pi}{8} \cdot 7 ~=~ \frac{49\pi}{8}\]

     We know that a circle’s circumference is twice its radius times \(\pi\). Look at the circular base of the cone, and use the circumference formula to find its radius. Remember \(L\) is the circumference of the cone’s base.

     \[L=2\pi r\]

     Divide both sides by \(2\pi\) and switch sides to get \(r=\frac{L}{2\pi}\).

     \[r ~=~ \frac{L}{2\pi} ~=~ \frac{\frac{49\pi}{8}}{2\pi}\]

     \[r ~=~ \frac{49}{16}\]

     To get the height of the cone, use Pythagorean theorem, which states a right triangle with legs \(a\) and \(b\) and hypotenuse \(c\) is constrained by \(a^2+b^2=c^2\).

     \[\left(\frac{49}{16}\right)^2 + h^2 = (7)^2\]

     Solve for \(h\).

     \[h = \sqrt{(7)^2-\left(\frac{49}{16}\right)^2}\]

     \[h\approx 6.2945289\]

     Find the ratio.

     \[\frac{h}{r} ~=~ \frac{\sqrt{(7)^2-\left(\frac{49}{16}\right)^2}}{\frac{49}{16}}\]

     \[\frac{h}{r} \approx 2.0553564\]

     Fancy stuff

     With some thought, you might convince yourself that this ratio \(\frac{h}{r}\) should be independent of the radius of the original circle. For example, if you measured the radius \(R\) in inches (instead of feet), the \(\frac{h}{r}\) ratio shouldn’t change. Or if you doubled \(R\), it seems \(\frac{h}{r}\) should be the same.

     By approaching this problem more algebraically, you might reach some interesting insights. For example, if we define \(f\) to be the fraction of the original circle.

     \[f = \frac{\theta}{2\pi}\]

     You can show that the ratio of radii also equals \(f\).

     \[\frac{r}{R} = f\]

     Substituting \(r=Rf\), you can generalize the formula to find \(h\) in terms of \(R\) and \(f\): \[h ~=~ \sqrt{R^2-r^2} ~=~ \sqrt{R^2-(fR)^2} ~=~ R\sqrt{1-f^2}\]

     Using that information, you can derive a simple formula for \(\frac{h}{r}\) that only depends on \(f\).